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Properties

Label 2-624-52.31-c3-0-13
Degree 22
Conductor 624624
Sign −0.477−0.878i-0.477 - 0.878i
Analytic cond. 36.817136.8171
Root an. cond. 6.067716.06771
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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

Λ(s)=(624s/2 ΓC(s) L(s)=((−0.477−0.878i) Λ‾(4−s)\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}
Λ(s)=(624s/2 ΓC(s+3/2) L(s)=((−0.477−0.878i) Λ‾(1−s)\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: 22
Conductor: 624624    =    24⋅3⋅132^{4} \cdot 3 \cdot 13
Sign: −0.477−0.878i-0.477 - 0.878i
Analytic conductor: 36.817136.8171
Root analytic conductor: 6.067716.06771
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ624(31,⋅)\chi_{624} (31, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 624, ( :3/2), −0.477−0.878i)(2,\ 624,\ (\ :3/2),\ -0.477 - 0.878i)

Particular Values

L(2)L(2) ≈\approx 1.5595113381.559511338
L(12)L(\frac12) ≈\approx 1.5595113381.559511338
L(52)L(\frac{5}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
3 1−3iT 1 - 3iT
13 1+(−44.4−14.7i)T 1 + (-44.4 - 14.7i)T
good5 1+(1.88−1.88i)T−125iT2 1 + (1.88 - 1.88i)T - 125iT^{2}
7 1+(9.36−9.36i)T−343iT2 1 + (9.36 - 9.36i)T - 343iT^{2}
11 1+(−29.2+29.2i)T−1.33e3iT2 1 + (-29.2 + 29.2i)T - 1.33e3iT^{2}
17 1−6.76iT−4.91e3T2 1 - 6.76iT - 4.91e3T^{2}
19 1+(−17.4−17.4i)T+6.85e3iT2 1 + (-17.4 - 17.4i)T + 6.85e3iT^{2}
23 1−123.T+1.21e4T2 1 - 123.T + 1.21e4T^{2}
29 1+15.8T+2.43e4T2 1 + 15.8T + 2.43e4T^{2}
31 1+(23.2+23.2i)T+2.97e4iT2 1 + (23.2 + 23.2i)T + 2.97e4iT^{2}
37 1+(−100.−100.i)T+5.06e4iT2 1 + (-100. - 100. i)T + 5.06e4iT^{2}
41 1+(291.−291.i)T−6.89e4iT2 1 + (291. - 291. i)T - 6.89e4iT^{2}
43 1+383.T+7.95e4T2 1 + 383.T + 7.95e4T^{2}
47 1+(143.−143.i)T−1.03e5iT2 1 + (143. - 143. i)T - 1.03e5iT^{2}
53 1−213.T+1.48e5T2 1 - 213.T + 1.48e5T^{2}
59 1+(34.0−34.0i)T−2.05e5iT2 1 + (34.0 - 34.0i)T - 2.05e5iT^{2}
61 1+389.T+2.26e5T2 1 + 389.T + 2.26e5T^{2}
67 1+(−166.−166.i)T+3.00e5iT2 1 + (-166. - 166. i)T + 3.00e5iT^{2}
71 1+(417.+417.i)T+3.57e5iT2 1 + (417. + 417. i)T + 3.57e5iT^{2}
73 1+(−409.−409.i)T+3.89e5iT2 1 + (-409. - 409. i)T + 3.89e5iT^{2}
79 1+1.04e3iT−4.93e5T2 1 + 1.04e3iT - 4.93e5T^{2}
83 1+(−75.5−75.5i)T+5.71e5iT2 1 + (-75.5 - 75.5i)T + 5.71e5iT^{2}
89 1+(−682.−682.i)T+7.04e5iT2 1 + (-682. - 682. i)T + 7.04e5iT^{2}
97 1+(−218.+218.i)T−9.12e5iT2 1 + (-218. + 218. i)T - 9.12e5iT^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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 ZZ-function along the critical line