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Properties

Label 2-616-616.293-c1-0-5
Degree $2$
Conductor $616$
Sign $-0.658 - 0.752i$
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  + (−1.18 − 0.776i)2-s + (0.794 + 1.83i)4-s + (−2.51 + 0.817i)7-s + (0.485 − 2.78i)8-s + (2.42 + 1.76i)9-s + (−3.17 − 0.964i)11-s + (3.60 + 0.987i)14-s + (−2.73 + 2.91i)16-s + (−1.5 − 3.96i)18-s + (3.00 + 3.60i)22-s − 7.50·23-s + (−1.54 + 4.75i)25-s + (−3.5 − 3.96i)28-s + (−1.42 − 4.37i)29-s + (5.5 − 1.32i)32-s + ⋯
L(s)  = 1  + (−0.835 − 0.548i)2-s + (0.397 + 0.917i)4-s + (−0.951 + 0.309i)7-s + (0.171 − 0.985i)8-s + (0.809 + 0.587i)9-s + (−0.956 − 0.290i)11-s + (0.964 + 0.263i)14-s + (−0.684 + 0.729i)16-s + (−0.353 − 0.935i)18-s + (0.639 + 0.768i)22-s − 1.56·23-s + (−0.309 + 0.951i)25-s + (−0.661 − 0.749i)28-s + (−0.264 − 0.812i)29-s + (0.972 − 0.233i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.658 - 0.752i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.658 - 0.752i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(616\)    =    \(2^{3} \cdot 7 \cdot 11\)
Sign: $-0.658 - 0.752i$
Analytic conductor: \(4.91878\)
Root analytic conductor: \(2.21783\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{616} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 616,\ (\ :1/2),\ -0.658 - 0.752i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0976796 + 0.215419i\)
\(L(\frac12)\) \(\approx\) \(0.0976796 + 0.215419i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.18 + 0.776i)T \)
7 \( 1 + (2.51 - 0.817i)T \)
11 \( 1 + (3.17 + 0.964i)T \)
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 \( 1 + 7.50T + 23T^{2} \)
29 \( 1 + (1.42 + 4.37i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-9.57 - 29.4i)T^{2} \)
37 \( 1 + (8.53 - 2.77i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 6.59T + 43T^{2} \)
47 \( 1 + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (8.48 - 11.6i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-18.8 + 58.0i)T^{2} \)
67 \( 1 - 15.1iT - 67T^{2} \)
71 \( 1 + (-7.95 + 5.78i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (4.78 - 6.58i)T + (-24.4 - 75.1i)T^{2} \)
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

−10.66814436368101217640603895635, −10.06528322883792330574979663720, −9.445790515871554617844699556522, −8.333396061937410749187110705339, −7.63910470338493932815394535663, −6.71141230593787969212847626531, −5.56494728297293120032521943029, −4.10487283575915551690027775543, −3.00758006713854004195453504411, −1.85984835638860032869709290658, 0.16021047466823987636356346468, 1.97596208865787983616628022619, 3.55557180477066514800372269758, 4.91310206593383177136049650107, 6.10177084704781423095898595573, 6.81158006250511834761996613089, 7.62629181487829125489163971276, 8.518528108253619582644971189622, 9.638372445262164973171046381909, 10.04644398022269287332696812576

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