Properties

Label 2-624-52.31-c3-0-13
Degree $2$
Conductor $624$
Sign $-0.477 - 0.878i$
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  + 3i·3-s + (−1.88 + 1.88i)5-s + (−9.36 + 9.36i)7-s − 9·9-s + (29.2 − 29.2i)11-s + (44.4 + 14.7i)13-s + (−5.65 − 5.65i)15-s + 6.76i·17-s + (17.4 + 17.4i)19-s + (−28.0 − 28.0i)21-s + 123.·23-s + 117. i·25-s − 27i·27-s − 15.8·29-s + (−23.2 − 23.2i)31-s + ⋯
L(s)  = 1  + 0.577i·3-s + (−0.168 + 0.168i)5-s + (−0.505 + 0.505i)7-s − 0.333·9-s + (0.801 − 0.801i)11-s + (0.949 + 0.314i)13-s + (−0.0974 − 0.0974i)15-s + 0.0964i·17-s + (0.210 + 0.210i)19-s + (−0.291 − 0.291i)21-s + 1.11·23-s + 0.943i·25-s − 0.192i·27-s − 0.101·29-s + (−0.134 − 0.134i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.477 - 0.878i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.477 - 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign: $-0.477 - 0.878i$
Analytic conductor: \(36.8171\)
Root analytic conductor: \(6.06771\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{624} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 624,\ (\ :3/2),\ -0.477 - 0.878i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.559511338\)
\(L(\frac12)\) \(\approx\) \(1.559511338\)
\(L(\frac{5}{2})\) 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 \( 1 - 3iT \)
13 \( 1 + (-44.4 - 14.7i)T \)
good5 \( 1 + (1.88 - 1.88i)T - 125iT^{2} \)
7 \( 1 + (9.36 - 9.36i)T - 343iT^{2} \)
11 \( 1 + (-29.2 + 29.2i)T - 1.33e3iT^{2} \)
17 \( 1 - 6.76iT - 4.91e3T^{2} \)
19 \( 1 + (-17.4 - 17.4i)T + 6.85e3iT^{2} \)
23 \( 1 - 123.T + 1.21e4T^{2} \)
29 \( 1 + 15.8T + 2.43e4T^{2} \)
31 \( 1 + (23.2 + 23.2i)T + 2.97e4iT^{2} \)
37 \( 1 + (-100. - 100. i)T + 5.06e4iT^{2} \)
41 \( 1 + (291. - 291. i)T - 6.89e4iT^{2} \)
43 \( 1 + 383.T + 7.95e4T^{2} \)
47 \( 1 + (143. - 143. i)T - 1.03e5iT^{2} \)
53 \( 1 - 213.T + 1.48e5T^{2} \)
59 \( 1 + (34.0 - 34.0i)T - 2.05e5iT^{2} \)
61 \( 1 + 389.T + 2.26e5T^{2} \)
67 \( 1 + (-166. - 166. i)T + 3.00e5iT^{2} \)
71 \( 1 + (417. + 417. i)T + 3.57e5iT^{2} \)
73 \( 1 + (-409. - 409. i)T + 3.89e5iT^{2} \)
79 \( 1 + 1.04e3iT - 4.93e5T^{2} \)
83 \( 1 + (-75.5 - 75.5i)T + 5.71e5iT^{2} \)
89 \( 1 + (-682. - 682. i)T + 7.04e5iT^{2} \)
97 \( 1 + (-218. + 218. i)T - 9.12e5iT^{2} \)
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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

−10.55678000707480241244910108210, −9.437782912526796299541530437698, −8.957515877736797767446649535860, −8.053955712906930888256427314687, −6.70393537556833387773020313825, −6.04568983670596202064206084385, −4.97465135225796362420286654139, −3.69554966474797597878808306816, −3.07269957885072638014893404229, −1.28978482137033217364516649320, 0.49607650023207895157858490022, 1.67396705879969409740543692357, 3.18229386312174790557140265417, 4.17835568247006181674055551269, 5.38718068907227824186951208890, 6.62511094261972450573217593906, 7.02594098304313096799259059503, 8.178441268681013681690692135102, 8.975977136861040316028991229643, 9.912882552952982851450082691465

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