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Properties

Label 2-3716-3716.367-c0-0-0
Degree $2$
Conductor $3716$
Sign $-0.981 - 0.191i$
Analytic cond. $1.85452$
Root an. cond. $1.36180$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.0270 + 0.999i)2-s + (−0.998 − 0.0541i)4-s + (1.47 + 1.25i)5-s + (0.0811 − 0.996i)8-s + (−0.999 + 0.0270i)9-s + (−1.29 + 1.44i)10-s + (−0.627 + 0.438i)13-s + (0.994 + 0.108i)16-s + (0.582 + 1.65i)17-s − i·18-s + (−1.40 − 1.33i)20-s + (0.447 + 2.73i)25-s + (−0.420 − 0.639i)26-s + (−0.266 − 1.95i)29-s + (−0.135 + 0.990i)32-s + ⋯
L(s)  = 1  + (−0.0270 + 0.999i)2-s + (−0.998 − 0.0541i)4-s + (1.47 + 1.25i)5-s + (0.0811 − 0.996i)8-s + (−0.999 + 0.0270i)9-s + (−1.29 + 1.44i)10-s + (−0.627 + 0.438i)13-s + (0.994 + 0.108i)16-s + (0.582 + 1.65i)17-s − i·18-s + (−1.40 − 1.33i)20-s + (0.447 + 2.73i)25-s + (−0.420 − 0.639i)26-s + (−0.266 − 1.95i)29-s + (−0.135 + 0.990i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3716 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.981 - 0.191i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3716 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.981 - 0.191i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3716\)    =    \(2^{2} \cdot 929\)
Sign: $-0.981 - 0.191i$
Analytic conductor: \(1.85452\)
Root analytic conductor: \(1.36180\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3716} (367, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3716,\ (\ :0),\ -0.981 - 0.191i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.204896032\)
\(L(\frac12)\) \(\approx\) \(1.204896032\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0270 - 0.999i)T \)
929 \( 1 + iT \)
good3 \( 1 + (0.999 - 0.0270i)T^{2} \)
5 \( 1 + (-1.47 - 1.25i)T + (0.161 + 0.986i)T^{2} \)
7 \( 1 + (0.444 - 0.895i)T^{2} \)
11 \( 1 + (0.762 - 0.647i)T^{2} \)
13 \( 1 + (0.627 - 0.438i)T + (0.344 - 0.938i)T^{2} \)
17 \( 1 + (-0.582 - 1.65i)T + (-0.779 + 0.626i)T^{2} \)
19 \( 1 + (-0.647 + 0.762i)T^{2} \)
23 \( 1 + (-0.0541 - 0.998i)T^{2} \)
29 \( 1 + (0.266 + 1.95i)T + (-0.963 + 0.267i)T^{2} \)
31 \( 1 + (-0.135 + 0.990i)T^{2} \)
37 \( 1 + (-0.211 + 0.961i)T + (-0.907 - 0.419i)T^{2} \)
41 \( 1 + (0.559 - 1.91i)T + (-0.842 - 0.538i)T^{2} \)
43 \( 1 + (0.999 - 0.0270i)T^{2} \)
47 \( 1 + (-0.605 - 0.796i)T^{2} \)
53 \( 1 + (1.83 - 0.300i)T + (0.947 - 0.319i)T^{2} \)
59 \( 1 + (0.395 + 0.918i)T^{2} \)
61 \( 1 + (-1.68 - 0.617i)T + (0.762 + 0.647i)T^{2} \)
67 \( 1 + (-0.538 - 0.842i)T^{2} \)
71 \( 1 + (-0.319 - 0.947i)T^{2} \)
73 \( 1 + (0.193 + 0.940i)T + (-0.918 + 0.395i)T^{2} \)
79 \( 1 + (0.583 - 0.812i)T^{2} \)
83 \( 1 + (0.605 + 0.796i)T^{2} \)
89 \( 1 + (-0.159 + 0.722i)T + (-0.907 - 0.419i)T^{2} \)
97 \( 1 + (0.0732 - 0.0351i)T + (0.626 - 0.779i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.069770017081807372353175254490, −8.121651460643384694182585648649, −7.55883721865204659117453923946, −6.56514160912857196092857433410, −6.06530822224716195931689214929, −5.77980466203475952052334358441, −4.75125551679732586036490416980, −3.62121619474606292455367838018, −2.73197283810083300643283302081, −1.74577473171647190784387239316, 0.68468288738285688734656371933, 1.78026704032801967326349278325, 2.62321636992756071183460429295, 3.39778363470037678178456512152, 4.88319712865564747747786031970, 5.15822715916796416356037566559, 5.64524982140290873307183186460, 6.80555482294958081898055407962, 7.976654556561968825249074213354, 8.732053651346530564167286134335

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