Ok, somehow it seems either my laravel files were broken or nginx REALLY wants a server_name in its config.

Final config:

server {
    listen 80;
    server_name example.dev;
    root /srv/www/public;

    index index.html index.htm index.php;

    charset utf-8;

    location / {
        try_files $uri $uri/ /index.php?$query_string;
    }

    location = /favicon.ico { access_log off; log_not_found off; }
    location = /robots.txt  { access_log off; log_not_found off; }

    sendfile off;

    client_max_body_size 100m;

    location ~ \.php$ {
        fastcgi_split_path_info ^(.+\.php)(/.+)$;
        fastcgi_pass unix:/run/php/php7.0-fpm.sock;
        fastcgi_index index.php;
        include fastcgi_params;
        fastcgi_param SCRIPT_FILENAME $document_root$fastcgi_script_name;
        fastcgi_intercept_errors off;
        fastcgi_buffer_size 16k;
        fastcgi_buffers 4 16k;
        fastcgi_connect_timeout 300;
        fastcgi_send_timeout 300;
        fastcgi_read_timeout 300;
    }

    location ~ /\.ht {
        deny all;
    }
}