L-function 2-1734-17.16-c1-0-34

• Label: 2-1734-17.16-c1-0-34
• Degree: $2$
• Conductor: $1734$
• Sign: $-0.970 + 0.242i$
• Analytic cond.: $13.8460$
• Root an. cond.: $3.72102$
• Motivic weight: $1$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $1$

Dirichlet series

L(s)  = 1

− 2^-s + i·3^-s + 4^-s − 2i·5^-s − i·6^-s − 8^-s − 9^-s + 2i·10^-s + 4i·11^-s + i·12^-s − 2·13^-s + 2·15^-s + 16^-s + 18^-s − 4·19^-s − 2i·20^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 1734 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.970 + 0.242i)\, \overline{\Lambda}(2-s) \end{aligned}$

Invariants

Degree:	$2$
Conductor:	$1734$    =    $2 \cdot 3 \cdot 17^{2}$
Sign:	$-0.970 + 0.242i$
Analytic conductor:	$13.8460$
Root analytic conductor:	$3.72102$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1734} (577, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$1$
Selberg data:	$(2,\ 1734,\ (\ :1/2),\ -0.970 + 0.242i)$

Particular Values

$L(1)$	$=$	$0$
$L(\frac{3}{2})$		not available

Euler product

   $L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$

	$p$	$F_p(T)$
bad	2	$1 + T$
	3	$1 - iT$
	17	$1$
good	5	$1 + 2iT - 5T^{2}$
	7	$1 - 7T^{2}$
	11	$1 - 4iT - 11T^{2}$
	13	$1 + 2T + 13T^{2}$
	19	$1 + 4T + 19T^{2}$
	23	$1 - 23T^{2}$
	29	$1 + 10iT - 29T^{2}$
	31	$1 - 8iT - 31T^{2}$
	37	$1 + 2iT - 37T^{2}$

Imaginary part of the first few zeros on the critical line

−8.933761392799839826898100758992, −8.433760468622332108854992008419, −7.48720674231698928069959600385, −6.72187031728737423164665189200, −5.62162483568493530275223343667, −4.74903973721194273586276255785, −4.10371494679489060674602657297, −2.66795200858653271220835373618, −1.61730430388947983317664254113, 0, 1.53145053578433420290544987660, 2.73718349619107287341725545787, 3.37270404614000534435414361942, 4.88089984248490371332633681887, 6.09997107027575999209746548857, 6.53071028041173935888121967469, 7.36640210656233403252744450253, 8.099035156557818667115623743402, 8.780335538845723747815883561169

Graph of the $Z$-function along the critical line