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Properties

Label 2-405-1.1-c1-0-1
Degree 22
Conductor 405405
Sign 11
Analytic cond. 3.233943.23394
Root an. cond. 1.798311.79831
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.51·2-s + 4.32·4-s − 5-s − 0.514·7-s − 5.83·8-s + 2.51·10-s + 3.32·11-s − 1.32·13-s + 1.29·14-s + 6.02·16-s + 3.32·17-s − 1.32·19-s − 4.32·20-s − 8.34·22-s − 4.12·23-s + 25-s + 3.32·26-s − 2.22·28-s + 1.38·29-s + 8.73·31-s − 3.48·32-s − 8.34·34-s + 0.514·35-s + 0.292·37-s + 3.32·38-s + 5.83·40-s + 11.3·41-s + ⋯
L(s)  = 1  − 1.77·2-s + 2.16·4-s − 0.447·5-s − 0.194·7-s − 2.06·8-s + 0.795·10-s + 1.00·11-s − 0.366·13-s + 0.345·14-s + 1.50·16-s + 0.805·17-s − 0.303·19-s − 0.966·20-s − 1.78·22-s − 0.860·23-s + 0.200·25-s + 0.651·26-s − 0.419·28-s + 0.257·29-s + 1.56·31-s − 0.616·32-s − 1.43·34-s + 0.0869·35-s + 0.0481·37-s + 0.538·38-s + 0.922·40-s + 1.77·41-s + ⋯

Functional equation

Λ(s)=(405s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(405s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 405405    =    34⋅53^{4} \cdot 5
Sign: 11
Analytic conductor: 3.233943.23394
Root analytic conductor: 1.798311.79831
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 405, ( :1/2), 1)(2,\ 405,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.56168013390.5616801339
L(12)L(\frac12) ≈\approx 0.56168013390.5616801339
L(32)L(\frac{3}{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)
bad3 1 1
5 1+T 1 + T
good2 1+2.51T+2T2 1 + 2.51T + 2T^{2}
7 1+0.514T+7T2 1 + 0.514T + 7T^{2}
11 1−3.32T+11T2 1 - 3.32T + 11T^{2}
13 1+1.32T+13T2 1 + 1.32T + 13T^{2}
17 1−3.32T+17T2 1 - 3.32T + 17T^{2}
19 1+1.32T+19T2 1 + 1.32T + 19T^{2}
23 1+4.12T+23T2 1 + 4.12T + 23T^{2}
29 1−1.38T+29T2 1 - 1.38T + 29T^{2}
31 1−8.73T+31T2 1 - 8.73T + 31T^{2}
37 1−0.292T+37T2 1 - 0.292T + 37T^{2}
41 1−11.3T+41T2 1 - 11.3T + 41T^{2}
43 1−10.3T+43T2 1 - 10.3T + 43T^{2}
47 1−4.86T+47T2 1 - 4.86T + 47T^{2}
53 1−5.02T+53T2 1 - 5.02T + 53T^{2}
59 1−5.02T+59T2 1 - 5.02T + 59T^{2}
61 1−7.34T+61T2 1 - 7.34T + 61T^{2}
67 1−9.44T+67T2 1 - 9.44T + 67T^{2}
71 1+8.99T+71T2 1 + 8.99T + 71T^{2}
73 1−6.05T+73T2 1 - 6.05T + 73T^{2}
79 1+8.05T+79T2 1 + 8.05T + 79T^{2}
83 1+1.54T+83T2 1 + 1.54T + 83T^{2}
89 1−3T+89T2 1 - 3T + 89T^{2}
97 1+12.2T+97T2 1 + 12.2T + 97T^{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

−11.04285306493371508566371080667, −10.07146441959455442372232616676, −9.476007814914580912086630166011, −8.533713877497509955171401547219, −7.79205900265118743413620976189, −6.91898632223280747240435368876, −5.98567514029642307861749193439, −4.11886575434530500089847515341, −2.55449225267992608826531435509, −0.954570366014384757362295281472, 0.954570366014384757362295281472, 2.55449225267992608826531435509, 4.11886575434530500089847515341, 5.98567514029642307861749193439, 6.91898632223280747240435368876, 7.79205900265118743413620976189, 8.533713877497509955171401547219, 9.476007814914580912086630166011, 10.07146441959455442372232616676, 11.04285306493371508566371080667

Graph of the ZZ-function along the critical line