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Properties

Label 2-616-616.237-c1-0-75
Degree $2$
Conductor $616$
Sign $0.724 + 0.688i$
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 + (−1.55 − 2.14i)7-s + (2.80 − 0.399i)8-s + (−0.927 − 2.85i)9-s + (3.13 − 1.08i)11-s + (−2.33 − 2.91i)14-s + (3.92 − 0.750i)16-s + (−1.5 − 3.96i)18-s + (4.35 − 1.74i)22-s + 3.36·23-s + (4.04 + 2.93i)25-s + (−3.49 − 3.96i)28-s + (−7.64 + 5.55i)29-s + (5.49 − 1.32i)32-s + ⋯
L(s)  = 1  + (0.998 − 0.0471i)2-s + (0.995 − 0.0942i)4-s + (−0.587 − 0.809i)7-s + (0.989 − 0.141i)8-s + (−0.309 − 0.951i)9-s + (0.945 − 0.326i)11-s + (−0.625 − 0.780i)14-s + (0.982 − 0.187i)16-s + (−0.353 − 0.935i)18-s + (0.928 − 0.371i)22-s + 0.700·23-s + (0.809 + 0.587i)25-s + (−0.661 − 0.749i)28-s + (−1.41 + 1.03i)29-s + (0.972 − 0.233i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.724 + 0.688i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.724 + 0.688i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(616\)    =    \(2^{3} \cdot 7 \cdot 11\)
Sign: $0.724 + 0.688i$
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: \((2,\ 616,\ (\ :1/2),\ 0.724 + 0.688i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.51064 - 1.00264i\)
\(L(\frac12)\) \(\approx\) \(2.51064 - 1.00264i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.55 + 2.14i)T \)
11 \( 1 + (-3.13 + 1.08i)T \)
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 \( 1 - 3.36T + 23T^{2} \)
29 \( 1 + (7.64 - 5.55i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (25.0 - 18.2i)T^{2} \)
37 \( 1 + (-2.95 - 4.07i)T + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 1.32T + 43T^{2} \)
47 \( 1 + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (12.1 - 3.94i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (49.3 + 35.8i)T^{2} \)
67 \( 1 + 8.70iT - 67T^{2} \)
71 \( 1 + (0.0272 - 0.0839i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-2.57 + 0.835i)T + (63.9 - 46.4i)T^{2} \)
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

−10.85717527671849373084729369061, −9.691133951107873211653156752032, −8.933793136871812751255013982814, −7.49695819594297172051749018280, −6.70651363790409886853230518500, −6.09194099575092465201169729555, −4.86559846890696460195582335471, −3.70397115195460934538886036899, −3.16579532068711233169901147647, −1.23800824273968008012504866683, 1.98835821246166548165274523673, 3.01570722530438491875059487635, 4.22042008021571495736961770357, 5.23963074050077352236374448096, 6.07694396841963548117003010859, 6.93086902120546072176205804729, 7.934672802362048661998047520277, 9.001702491718435542576365558487, 9.952959627093119227928587199943, 11.06543913552384999083975639908

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