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Properties

Label 2-624-13.3-c3-0-16
Degree $2$
Conductor $624$
Sign $0.969 + 0.243i$
Analytic cond. $36.8171$
Root an. cond. $6.06771$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.5 − 2.59i)3-s − 5.85·5-s + (−12.0 + 20.8i)7-s + (−4.5 + 7.79i)9-s + (−16.9 − 29.3i)11-s + (−40.8 − 23.0i)13-s + (8.78 + 15.2i)15-s + (24.6 − 42.7i)17-s + (−38.4 + 66.5i)19-s + 72.3·21-s + (3.14 + 5.44i)23-s − 90.6·25-s + 27·27-s + (−50.4 − 87.4i)29-s + 307.·31-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s − 0.524·5-s + (−0.651 + 1.12i)7-s + (−0.166 + 0.288i)9-s + (−0.464 − 0.804i)11-s + (−0.870 − 0.492i)13-s + (0.151 + 0.262i)15-s + (0.352 − 0.610i)17-s + (−0.463 + 0.803i)19-s + 0.752·21-s + (0.0285 + 0.0493i)23-s − 0.725·25-s + 0.192·27-s + (−0.323 − 0.560i)29-s + 1.78·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 + 0.243i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.969 + 0.243i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $0.969 + 0.243i$
Analytic conductor: \(36.8171\)
Root analytic conductor: \(6.06771\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{624} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 624,\ (\ :3/2),\ 0.969 + 0.243i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.9860116194\)
\(L(\frac12)\) \(\approx\) \(0.9860116194\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (1.5 + 2.59i)T \)
13 \( 1 + (40.8 + 23.0i)T \)
good5 \( 1 + 5.85T + 125T^{2} \)
7 \( 1 + (12.0 - 20.8i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (16.9 + 29.3i)T + (-665.5 + 1.15e3i)T^{2} \)
17 \( 1 + (-24.6 + 42.7i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (38.4 - 66.5i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-3.14 - 5.44i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (50.4 + 87.4i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 - 307.T + 2.97e4T^{2} \)
37 \( 1 + (-38.0 - 65.8i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-257. - 445. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (134. - 232. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 - 460.T + 1.03e5T^{2} \)
53 \( 1 - 67.8T + 1.48e5T^{2} \)
59 \( 1 + (-12.6 + 21.8i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-294. + 509. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (502. + 869. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (-447. + 775. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 - 968.T + 3.89e5T^{2} \)
79 \( 1 - 119.T + 4.93e5T^{2} \)
83 \( 1 + 480.T + 5.71e5T^{2} \)
89 \( 1 + (542. + 940. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-8.32 + 14.4i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−10.10474018197748958444467263643, −9.360034373254255252830282334911, −8.127652412483624973845802007602, −7.77969567495578221841079248705, −6.37694848500169055515172788650, −5.81149750178565850870585564419, −4.75281350597124453317250769246, −3.22628219779038945621431401184, −2.39114714793762086770552010928, −0.56747296023011533067270233222, 0.57520323370447319176564832965, 2.46473780422317324955975719071, 3.90664315494120613216583277190, 4.40563173509966278832107466886, 5.59582843803388456876978851752, 6.93103330328553194902594284074, 7.32920041736910165971342625777, 8.527373875772839462156188273990, 9.651189188908419909933836621259, 10.22437552256942025905667682475

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