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Properties

Label 2-7605-1.1-c1-0-32
Degree 22
Conductor 76057605
Sign 11
Analytic cond. 60.726260.7262
Root an. cond. 7.792707.79270
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  − 1.56·2-s + 0.438·4-s + 5-s − 0.561·7-s + 2.43·8-s − 1.56·10-s + 1.43·11-s + 0.876·14-s − 4.68·16-s − 4.56·17-s − 7.12·19-s + 0.438·20-s − 2.24·22-s + 1.43·23-s + 25-s − 0.246·28-s + 8.24·29-s − 6·31-s + 2.43·32-s + 7.12·34-s − 0.561·35-s − 1.43·37-s + 11.1·38-s + 2.43·40-s + 4.56·41-s + 1.12·43-s + 0.630·44-s + ⋯
L(s)  = 1  − 1.10·2-s + 0.219·4-s + 0.447·5-s − 0.212·7-s + 0.862·8-s − 0.493·10-s + 0.433·11-s + 0.234·14-s − 1.17·16-s − 1.10·17-s − 1.63·19-s + 0.0980·20-s − 0.478·22-s + 0.299·23-s + 0.200·25-s − 0.0465·28-s + 1.53·29-s − 1.07·31-s + 0.431·32-s + 1.22·34-s − 0.0949·35-s − 0.236·37-s + 1.80·38-s + 0.385·40-s + 0.712·41-s + 0.171·43-s + 0.0950·44-s + ⋯

Functional equation

Λ(s)=(7605s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(7605s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 76057605    =    32⋅5⋅1323^{2} \cdot 5 \cdot 13^{2}
Sign: 11
Analytic conductor: 60.726260.7262
Root analytic conductor: 7.792707.79270
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 7605, ( :1/2), 1)(2,\ 7605,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.78295496050.7829549605
L(12)L(\frac12) ≈\approx 0.78295496050.7829549605
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
13 1 1
good2 1+1.56T+2T2 1 + 1.56T + 2T^{2}
7 1+0.561T+7T2 1 + 0.561T + 7T^{2}
11 1−1.43T+11T2 1 - 1.43T + 11T^{2}
17 1+4.56T+17T2 1 + 4.56T + 17T^{2}
19 1+7.12T+19T2 1 + 7.12T + 19T^{2}
23 1−1.43T+23T2 1 - 1.43T + 23T^{2}
29 1−8.24T+29T2 1 - 8.24T + 29T^{2}
31 1+6T+31T2 1 + 6T + 31T^{2}
37 1+1.43T+37T2 1 + 1.43T + 37T^{2}
41 1−4.56T+41T2 1 - 4.56T + 41T^{2}
43 1−1.12T+43T2 1 - 1.12T + 43T^{2}
47 1+4T+47T2 1 + 4T + 47T^{2}
53 1+10.8T+53T2 1 + 10.8T + 53T^{2}
59 1−4T+59T2 1 - 4T + 59T^{2}
61 1+12.5T+61T2 1 + 12.5T + 61T^{2}
67 1−11.1T+67T2 1 - 11.1T + 67T^{2}
71 1−3.68T+71T2 1 - 3.68T + 71T^{2}
73 1+2.87T+73T2 1 + 2.87T + 73T^{2}
79 1+1.43T+79T2 1 + 1.43T + 79T^{2}
83 1−11.3T+83T2 1 - 11.3T + 83T^{2}
89 1−13.6T+89T2 1 - 13.6T + 89T^{2}
97 1−13.9T+97T2 1 - 13.9T + 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

−8.132440212592810593275888644316, −7.25205233852613308409661805611, −6.53151885810136376711864138728, −6.16741889807198677992386620483, −4.85762674651273764162901638050, −4.49649604719643185178212294231, −3.47583711628516405329842142870, −2.31190517990692117373700324639, −1.68939782714197712377292569486, −0.52657234839112931251656539288, 0.52657234839112931251656539288, 1.68939782714197712377292569486, 2.31190517990692117373700324639, 3.47583711628516405329842142870, 4.49649604719643185178212294231, 4.85762674651273764162901638050, 6.16741889807198677992386620483, 6.53151885810136376711864138728, 7.25205233852613308409661805611, 8.132440212592810593275888644316

Graph of the ZZ-function along the critical line