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Properties

Label 2-616-616.237-c1-0-38
Degree $2$
Conductor $616$
Sign $0.470 + 0.882i$
Analytic cond. $4.91878$
Root an. cond. $2.21783$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.10 − 0.884i)2-s + (0.435 + 1.95i)4-s + (1.55 + 2.14i)7-s + (1.24 − 2.53i)8-s + (−0.927 − 2.85i)9-s + (−1.89 − 2.71i)11-s + (0.176 − 3.73i)14-s + (−3.61 + 1.70i)16-s + (−1.50 + 3.96i)18-s + (−0.310 + 4.68i)22-s + 9.58·23-s + (4.04 + 2.93i)25-s + (−3.50 + 3.96i)28-s + (8.64 − 6.27i)29-s + (5.49 + 1.32i)32-s + ⋯
L(s)  = 1  + (−0.780 − 0.625i)2-s + (0.217 + 0.975i)4-s + (0.587 + 0.809i)7-s + (0.440 − 0.897i)8-s + (−0.309 − 0.951i)9-s + (−0.572 − 0.820i)11-s + (0.0471 − 0.998i)14-s + (−0.904 + 0.425i)16-s + (−0.353 + 0.935i)18-s + (−0.0661 + 0.997i)22-s + 1.99·23-s + (0.809 + 0.587i)25-s + (−0.661 + 0.749i)28-s + (1.60 − 1.16i)29-s + (0.972 + 0.233i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.470 + 0.882i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.470 + 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(616\)    =    \(2^{3} \cdot 7 \cdot 11\)
Sign: $0.470 + 0.882i$
Analytic conductor: \(4.91878\)
Root analytic conductor: \(2.21783\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{616} (237, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 616,\ (\ :1/2),\ 0.470 + 0.882i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.875321 - 0.525337i\)
\(L(\frac12)\) \(\approx\) \(0.875321 - 0.525337i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.10 + 0.884i)T \)
7 \( 1 + (-1.55 - 2.14i)T \)
11 \( 1 + (1.89 + 2.71i)T \)
good3 \( 1 + (0.927 + 2.85i)T^{2} \)
5 \( 1 + (-4.04 - 2.93i)T^{2} \)
13 \( 1 + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-5.87 - 18.0i)T^{2} \)
23 \( 1 - 9.58T + 23T^{2} \)
29 \( 1 + (-8.64 + 6.27i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (25.0 - 18.2i)T^{2} \)
37 \( 1 + (7.10 + 9.77i)T + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 8.74T + 43T^{2} \)
47 \( 1 + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (5.93 - 1.92i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (49.3 + 35.8i)T^{2} \)
67 \( 1 - 1.10iT - 67T^{2} \)
71 \( 1 + (-3.08 + 9.48i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-11.9 + 3.86i)T + (63.9 - 46.4i)T^{2} \)
83 \( 1 + (67.1 + 48.7i)T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 + (78.4 - 57.0i)T^{2} \)
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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

−10.72423924476134321903667911317, −9.362030398028871418662237006258, −8.895876612672945765850090971498, −8.167639657142234671664251567324, −7.10547564384374543865195095364, −6.02039980943023621456536720280, −4.85325149551856566442292923613, −3.37612197953265889409702639944, −2.54522835830713063402408451653, −0.875232867518650919835395256217, 1.25505444065801953126887395828, 2.70823442000404898402370401055, 4.83915614281977566517754251713, 5.02784442255600638775977001447, 6.67692074048189050534188335177, 7.26829659583019788667380452948, 8.153603226090073121174209275084, 8.796880957152601427320821166006, 10.01910011163051096056514282426, 10.65657103101591700945416107921

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