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Properties

Label 2-616-616.349-c1-0-22
Degree $2$
Conductor $616$
Sign $0.966 - 0.258i$
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  + (−1.41 − 0.0667i)2-s + (1.99 + 0.188i)4-s + (2.51 + 0.817i)7-s + (−2.80 − 0.399i)8-s + (2.42 − 1.76i)9-s + (0.0629 + 3.31i)11-s + (−3.49 − 1.32i)14-s + (3.92 + 0.750i)16-s + (−3.54 + 2.32i)18-s + (0.132 − 4.68i)22-s + 2.56·23-s + (−1.54 − 4.75i)25-s + (4.85 + 2.10i)28-s + (−2.42 + 7.45i)29-s + (−5.49 − 1.32i)32-s + ⋯
L(s)  = 1  + (−0.998 − 0.0471i)2-s + (0.995 + 0.0942i)4-s + (0.951 + 0.309i)7-s + (−0.989 − 0.141i)8-s + (0.809 − 0.587i)9-s + (0.0189 + 0.999i)11-s + (−0.935 − 0.353i)14-s + (0.982 + 0.187i)16-s + (−0.835 + 0.548i)18-s + (0.0282 − 0.999i)22-s + 0.533·23-s + (−0.309 − 0.951i)25-s + (0.917 + 0.397i)28-s + (−0.449 + 1.38i)29-s + (−0.972 − 0.233i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 - 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.966 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(616\)    =    \(2^{3} \cdot 7 \cdot 11\)
Sign: $0.966 - 0.258i$
Analytic conductor: \(4.91878\)
Root analytic conductor: \(2.21783\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{616} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 616,\ (\ :1/2),\ 0.966 - 0.258i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.12188 + 0.147263i\)
\(L(\frac12)\) \(\approx\) \(1.12188 + 0.147263i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 + 0.0667i)T \)
7 \( 1 + (-2.51 - 0.817i)T \)
11 \( 1 + (-0.0629 - 3.31i)T \)
good3 \( 1 + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (1.54 + 4.75i)T^{2} \)
13 \( 1 + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (15.3 - 11.1i)T^{2} \)
23 \( 1 - 2.56T + 23T^{2} \)
29 \( 1 + (2.42 - 7.45i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-9.57 + 29.4i)T^{2} \)
37 \( 1 + (-2.31 - 0.752i)T + (29.9 + 21.7i)T^{2} \)
41 \( 1 + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 12.8T + 43T^{2} \)
47 \( 1 + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (1.57 + 2.17i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-18.8 - 58.0i)T^{2} \)
67 \( 1 + 10.4iT - 67T^{2} \)
71 \( 1 + (-12.9 - 9.43i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-10.3 - 14.1i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (-25.6 - 78.9i)T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 + (-29.9 + 92.2i)T^{2} \)
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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

−10.59459792693574355085388196565, −9.666946188340865104923204099506, −9.052221464037930585449584357448, −8.039488187521448267794163470229, −7.28865227555266939793644954704, −6.48957758619184686823331521399, −5.19929675054873352037202352077, −4.00779075136514001642507305224, −2.41028149941987957331238616768, −1.28367317085303204739311127376, 1.08059428793675060582167242276, 2.33325643845703054816974830788, 3.88203070682436199287923000644, 5.20982111582213259137152089080, 6.23243516437002735464504455127, 7.48655354328485902187439221977, 7.82078472145796158888212708380, 8.848907693050504054944360836224, 9.665355481969926666404809874131, 10.72135846055115210961083841146

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