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Properties

Label 2-624-13.3-c3-0-9
Degree $2$
Conductor $624$
Sign $-0.653 - 0.756i$
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 − 3.53·5-s + (0.928 − 1.60i)7-s + (−4.5 + 7.79i)9-s + (21.0 + 36.4i)11-s + (20.1 + 42.3i)13-s + (−5.30 − 9.18i)15-s + (5.33 − 9.23i)17-s + (35.8 − 62.0i)19-s + 5.57·21-s + (−47.1 − 81.7i)23-s − 112.·25-s − 27·27-s + (50.0 + 86.6i)29-s − 118.·31-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s − 0.316·5-s + (0.0501 − 0.0868i)7-s + (−0.166 + 0.288i)9-s + (0.576 + 0.998i)11-s + (0.429 + 0.903i)13-s + (−0.0912 − 0.158i)15-s + (0.0760 − 0.131i)17-s + (0.432 − 0.749i)19-s + 0.0579·21-s + (−0.427 − 0.740i)23-s − 0.900·25-s − 0.192·27-s + (0.320 + 0.554i)29-s − 0.684·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.653 - 0.756i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.653 - 0.756i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $-0.653 - 0.756i$
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: \((2,\ 624,\ (\ :3/2),\ -0.653 - 0.756i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.526626975\)
\(L(\frac12)\) \(\approx\) \(1.526626975\)
\(L(\frac{5}{2})\) 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 \( 1 + (-20.1 - 42.3i)T \)
good5 \( 1 + 3.53T + 125T^{2} \)
7 \( 1 + (-0.928 + 1.60i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (-21.0 - 36.4i)T + (-665.5 + 1.15e3i)T^{2} \)
17 \( 1 + (-5.33 + 9.23i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-35.8 + 62.0i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (47.1 + 81.7i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-50.0 - 86.6i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + 118.T + 2.97e4T^{2} \)
37 \( 1 + (-69.2 - 119. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-252. - 437. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-36.2 + 62.7i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + 228.T + 1.03e5T^{2} \)
53 \( 1 + 267.T + 1.48e5T^{2} \)
59 \( 1 + (183. - 317. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (415. - 718. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (120. + 208. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (7.29 - 12.6i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + 955.T + 3.89e5T^{2} \)
79 \( 1 + 711.T + 4.93e5T^{2} \)
83 \( 1 - 111.T + 5.71e5T^{2} \)
89 \( 1 + (-452. - 782. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (210. - 365. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−10.45662680410255871346659982575, −9.516424721900823801011807188579, −8.998439543408944075763082864646, −7.893350193554201845700498974144, −7.04642584877257019256094509201, −6.06313451587399200186380158037, −4.64436093994287163180691710648, −4.14595313358611862156994073778, −2.84492265886952299542418113987, −1.48475091916213921188846540179, 0.43045433430697073384094015916, 1.71864471724109870865428601606, 3.21404752268462269580092667585, 3.93635499920047025263813732965, 5.59864694460454635554883464154, 6.13656026925050867626900981337, 7.46523264044895478455960153214, 8.044751545605028097330795081027, 8.884343411824821551524757551057, 9.819686382192488507800627225652

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