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Properties

Label 2-624-12.11-c3-0-7
Degree $2$
Conductor $624$
Sign $-0.618 + 0.785i$
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  + (1.92 + 4.82i)3-s + 18.6i·5-s − 4.79i·7-s + (−19.5 + 18.5i)9-s − 18.4·11-s − 13·13-s + (−89.9 + 35.9i)15-s + 42.4i·17-s − 58.4i·19-s + (23.1 − 9.23i)21-s − 20.6·23-s − 222.·25-s + (−127. − 58.6i)27-s − 65.8i·29-s + 129. i·31-s + ⋯
L(s)  = 1  + (0.370 + 0.928i)3-s + 1.66i·5-s − 0.258i·7-s + (−0.725 + 0.688i)9-s − 0.505·11-s − 0.277·13-s + (−1.54 + 0.618i)15-s + 0.605i·17-s − 0.705i·19-s + (0.240 − 0.0959i)21-s − 0.186·23-s − 1.78·25-s + (−0.908 − 0.417i)27-s − 0.421i·29-s + 0.748i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.618 + 0.785i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.618 + 0.785i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $-0.618 + 0.785i$
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: \((2,\ 624,\ (\ :3/2),\ -0.618 + 0.785i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.9016305200\)
\(L(\frac12)\) \(\approx\) \(0.9016305200\)
\(L(\frac{5}{2})\) 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 \( 1 + (-1.92 - 4.82i)T \)
13 \( 1 + 13T \)
good5 \( 1 - 18.6iT - 125T^{2} \)
7 \( 1 + 4.79iT - 343T^{2} \)
11 \( 1 + 18.4T + 1.33e3T^{2} \)
17 \( 1 - 42.4iT - 4.91e3T^{2} \)
19 \( 1 + 58.4iT - 6.85e3T^{2} \)
23 \( 1 + 20.6T + 1.21e4T^{2} \)
29 \( 1 + 65.8iT - 2.43e4T^{2} \)
31 \( 1 - 129. iT - 2.97e4T^{2} \)
37 \( 1 + 280.T + 5.06e4T^{2} \)
41 \( 1 + 49.8iT - 6.89e4T^{2} \)
43 \( 1 - 336. iT - 7.95e4T^{2} \)
47 \( 1 - 482.T + 1.03e5T^{2} \)
53 \( 1 + 256. iT - 1.48e5T^{2} \)
59 \( 1 + 96.1T + 2.05e5T^{2} \)
61 \( 1 + 275.T + 2.26e5T^{2} \)
67 \( 1 + 912. iT - 3.00e5T^{2} \)
71 \( 1 - 842.T + 3.57e5T^{2} \)
73 \( 1 + 851.T + 3.89e5T^{2} \)
79 \( 1 - 305. iT - 4.93e5T^{2} \)
83 \( 1 - 394.T + 5.71e5T^{2} \)
89 \( 1 - 1.35e3iT - 7.04e5T^{2} \)
97 \( 1 - 261.T + 9.12e5T^{2} \)
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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

−10.65399811953720297990789748261, −10.13684724437901491152706601960, −9.200695877800232147522541947602, −8.096879101250444998255460588080, −7.26613625094839999110783026361, −6.34293507908729015909822428809, −5.22733193060623562613784522440, −4.02340282563261573850654717209, −3.12892802222378269550616620052, −2.29725216455347700948704489997, 0.24119068026179436863371928621, 1.36002558899392762670704675039, 2.47264953233651532728917553068, 3.95972915843902528067601770078, 5.19756922026556717137233442521, 5.83099754869477433387687138390, 7.19590319017204658363999993188, 7.960831720387774169335902692398, 8.771053876962845925658829989565, 9.264373332141908730667928948275

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