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Properties

Label 2-624-12.11-c3-0-15
Degree $2$
Conductor $624$
Sign $0.165 - 0.986i$
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  + (0.861 − 5.12i)3-s + 19.6i·5-s + 0.649i·7-s + (−25.5 − 8.82i)9-s + 9.64·11-s + 13·13-s + (100. + 16.8i)15-s − 48.1i·17-s − 103. i·19-s + (3.33 + 0.559i)21-s + 170.·23-s − 259.·25-s + (−67.2 + 123. i)27-s + 218. i·29-s + 291. i·31-s + ⋯
L(s)  = 1  + (0.165 − 0.986i)3-s + 1.75i·5-s + 0.0350i·7-s + (−0.945 − 0.326i)9-s + 0.264·11-s + 0.277·13-s + (1.72 + 0.290i)15-s − 0.686i·17-s − 1.25i·19-s + (0.0346 + 0.00581i)21-s + 1.54·23-s − 2.07·25-s + (−0.479 + 0.877i)27-s + 1.40i·29-s + 1.68i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.165 - 0.986i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.165 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $0.165 - 0.986i$
Analytic conductor: \(36.8171\)
Root analytic conductor: \(6.06771\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{624} (287, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 624,\ (\ :3/2),\ 0.165 - 0.986i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.552494298\)
\(L(\frac12)\) \(\approx\) \(1.552494298\)
\(L(\frac{5}{2})\) 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 \( 1 + (-0.861 + 5.12i)T \)
13 \( 1 - 13T \)
good5 \( 1 - 19.6iT - 125T^{2} \)
7 \( 1 - 0.649iT - 343T^{2} \)
11 \( 1 - 9.64T + 1.33e3T^{2} \)
17 \( 1 + 48.1iT - 4.91e3T^{2} \)
19 \( 1 + 103. iT - 6.85e3T^{2} \)
23 \( 1 - 170.T + 1.21e4T^{2} \)
29 \( 1 - 218. iT - 2.43e4T^{2} \)
31 \( 1 - 291. iT - 2.97e4T^{2} \)
37 \( 1 + 217.T + 5.06e4T^{2} \)
41 \( 1 - 353. iT - 6.89e4T^{2} \)
43 \( 1 - 426. iT - 7.95e4T^{2} \)
47 \( 1 + 105.T + 1.03e5T^{2} \)
53 \( 1 - 286. iT - 1.48e5T^{2} \)
59 \( 1 + 742.T + 2.05e5T^{2} \)
61 \( 1 + 238.T + 2.26e5T^{2} \)
67 \( 1 - 139. iT - 3.00e5T^{2} \)
71 \( 1 - 88.4T + 3.57e5T^{2} \)
73 \( 1 - 131.T + 3.89e5T^{2} \)
79 \( 1 - 907. iT - 4.93e5T^{2} \)
83 \( 1 + 245.T + 5.71e5T^{2} \)
89 \( 1 + 771. iT - 7.04e5T^{2} \)
97 \( 1 + 777.T + 9.12e5T^{2} \)
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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

−10.80800981182468870133046003627, −9.431508611244753840387117278318, −8.620527036429128709014126349175, −7.34775301865745633863059092092, −6.95146305949408390741967975472, −6.31747199123322403315132877796, −5.00502534804782143492820328497, −3.13496887786614137807848752184, −2.86231930087723843457588687989, −1.31679265758833469205848007962, 0.45673689556929798309923633616, 1.88938822421449671270204562637, 3.67813225998312111810319686590, 4.33526465356960794034362616342, 5.32140309084408417924473727275, 5.98691755405171836767028103151, 7.71561290283642938834112300841, 8.551152252130169599195812338391, 9.071566064065747030690835582370, 9.855799778243355254393474587717

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