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Properties

Label 2-2523-87.62-c0-0-3
Degree 22
Conductor 25232523
Sign 0.620−0.784i0.620 - 0.784i
Analytic cond. 1.259141.25914
Root an. cond. 1.122111.12211
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.222 + 0.974i)2-s + (0.900 − 0.433i)3-s + (0.222 + 0.974i)6-s + (0.900 − 0.433i)7-s + (−0.623 + 0.781i)8-s + (0.623 − 0.781i)9-s + (0.623 + 0.781i)11-s + (−0.623 − 0.781i)13-s + (0.222 + 0.974i)14-s + (−0.623 − 0.781i)16-s + 17-s + (0.623 + 0.781i)18-s + (0.623 − 0.781i)21-s + (−0.900 + 0.433i)22-s + (−0.222 + 0.974i)24-s + (−0.900 − 0.433i)25-s + ⋯
L(s)  = 1  + (−0.222 + 0.974i)2-s + (0.900 − 0.433i)3-s + (0.222 + 0.974i)6-s + (0.900 − 0.433i)7-s + (−0.623 + 0.781i)8-s + (0.623 − 0.781i)9-s + (0.623 + 0.781i)11-s + (−0.623 − 0.781i)13-s + (0.222 + 0.974i)14-s + (−0.623 − 0.781i)16-s + 17-s + (0.623 + 0.781i)18-s + (0.623 − 0.781i)21-s + (−0.900 + 0.433i)22-s + (−0.222 + 0.974i)24-s + (−0.900 − 0.433i)25-s + ⋯

Functional equation

Λ(s)=(2523s/2 ΓC(s) L(s)=((0.620−0.784i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2523 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.620 - 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2523s/2 ΓC(s) L(s)=((0.620−0.784i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2523 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.620 - 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 25232523    =    3⋅2923 \cdot 29^{2}
Sign: 0.620−0.784i0.620 - 0.784i
Analytic conductor: 1.259141.25914
Root analytic conductor: 1.122111.12211
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2523(236,⋅)\chi_{2523} (236, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2523, ( :0), 0.620−0.784i)(2,\ 2523,\ (\ :0),\ 0.620 - 0.784i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.7798947561.779894756
L(12)L(\frac12) ≈\approx 1.7798947561.779894756
L(1)L(1) 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)
bad3 1+(−0.900+0.433i)T 1 + (-0.900 + 0.433i)T
29 1 1
good2 1+(0.222−0.974i)T+(−0.900−0.433i)T2 1 + (0.222 - 0.974i)T + (-0.900 - 0.433i)T^{2}
5 1+(0.900+0.433i)T2 1 + (0.900 + 0.433i)T^{2}
7 1+(−0.900+0.433i)T+(0.623−0.781i)T2 1 + (-0.900 + 0.433i)T + (0.623 - 0.781i)T^{2}
11 1+(−0.623−0.781i)T+(−0.222+0.974i)T2 1 + (-0.623 - 0.781i)T + (-0.222 + 0.974i)T^{2}
13 1+(0.623+0.781i)T+(−0.222+0.974i)T2 1 + (0.623 + 0.781i)T + (-0.222 + 0.974i)T^{2}
17 1−T+T2 1 - T + T^{2}
19 1+(−0.623−0.781i)T2 1 + (-0.623 - 0.781i)T^{2}
23 1+(0.900−0.433i)T2 1 + (0.900 - 0.433i)T^{2}
31 1+(0.900+0.433i)T2 1 + (0.900 + 0.433i)T^{2}
37 1+(0.222+0.974i)T2 1 + (0.222 + 0.974i)T^{2}
41 1+2T+T2 1 + 2T + T^{2}
43 1+(0.900−0.433i)T2 1 + (0.900 - 0.433i)T^{2}
47 1+(−0.623−0.781i)T+(−0.222+0.974i)T2 1 + (-0.623 - 0.781i)T + (-0.222 + 0.974i)T^{2}
53 1+(0.900+0.433i)T2 1 + (0.900 + 0.433i)T^{2}
59 1−T2 1 - T^{2}
61 1+(−0.623+0.781i)T2 1 + (-0.623 + 0.781i)T^{2}
67 1+(0.623−0.781i)T+(−0.222−0.974i)T2 1 + (0.623 - 0.781i)T + (-0.222 - 0.974i)T^{2}
71 1+(0.222−0.974i)T2 1 + (0.222 - 0.974i)T^{2}
73 1+(0.900−0.433i)T2 1 + (0.900 - 0.433i)T^{2}
79 1+(0.222+0.974i)T2 1 + (0.222 + 0.974i)T^{2}
83 1+(−0.623−0.781i)T2 1 + (-0.623 - 0.781i)T^{2}
89 1+(0.222−0.974i)T+(−0.900−0.433i)T2 1 + (0.222 - 0.974i)T + (-0.900 - 0.433i)T^{2}
97 1+(−0.623−0.781i)T2 1 + (-0.623 - 0.781i)T^{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

−8.895257649294070106431496744333, −8.115360628252380384412767805033, −7.69820461110891227431018528694, −7.18934936123385080128043644076, −6.40868037634175113343799721545, −5.44804104239928780293585493216, −4.54058348087856805188807201916, −3.48980652330490030812847950301, −2.45922063554629742235526816478, −1.46367105230593812241287451839, 1.50422041062164268881109885976, 2.12464774612047355958199990126, 3.19619537241012289496920419142, 3.80782940858434823116273448153, 4.83393174029117698606661653496, 5.74942279839492074761531418050, 6.83235317800709142146737978998, 7.67810613725052889893878550109, 8.529666158509491563763641689339, 9.062984084818235692144053012169

Graph of the ZZ-function along the critical line