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Properties

Label 2-7605-1.1-c1-0-153
Degree 22
Conductor 76057605
Sign −1-1
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 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.73·2-s + 5.46·4-s + 5-s − 1.73·7-s − 9.46·8-s − 2.73·10-s − 2·11-s + 4.73·14-s + 14.9·16-s + 2.73·17-s + 7.46·19-s + 5.46·20-s + 5.46·22-s − 2·23-s + 25-s − 9.46·28-s + 6.73·29-s + 2.46·31-s − 21.8·32-s − 7.46·34-s − 1.73·35-s − 10.3·37-s − 20.3·38-s − 9.46·40-s − 7.26·41-s + 1.19·43-s − 10.9·44-s + ⋯
L(s)  = 1  − 1.93·2-s + 2.73·4-s + 0.447·5-s − 0.654·7-s − 3.34·8-s − 0.863·10-s − 0.603·11-s + 1.26·14-s + 3.73·16-s + 0.662·17-s + 1.71·19-s + 1.22·20-s + 1.16·22-s − 0.417·23-s + 0.200·25-s − 1.78·28-s + 1.25·29-s + 0.442·31-s − 3.86·32-s − 1.28·34-s − 0.292·35-s − 1.70·37-s − 3.30·38-s − 1.49·40-s − 1.13·41-s + 0.182·43-s − 1.64·44-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: −1-1
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: 11
Selberg data: (2, 7605, ( :1/2), −1)(2,\ 7605,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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+2.73T+2T2 1 + 2.73T + 2T^{2}
7 1+1.73T+7T2 1 + 1.73T + 7T^{2}
11 1+2T+11T2 1 + 2T + 11T^{2}
17 1−2.73T+17T2 1 - 2.73T + 17T^{2}
19 1−7.46T+19T2 1 - 7.46T + 19T^{2}
23 1+2T+23T2 1 + 2T + 23T^{2}
29 1−6.73T+29T2 1 - 6.73T + 29T^{2}
31 1−2.46T+31T2 1 - 2.46T + 31T^{2}
37 1+10.3T+37T2 1 + 10.3T + 37T^{2}
41 1+7.26T+41T2 1 + 7.26T + 41T^{2}
43 1−1.19T+43T2 1 - 1.19T + 43T^{2}
47 1+10.1T+47T2 1 + 10.1T + 47T^{2}
53 1−2.53T+53T2 1 - 2.53T + 53T^{2}
59 1−5.66T+59T2 1 - 5.66T + 59T^{2}
61 1+15.3T+61T2 1 + 15.3T + 61T^{2}
67 1+1.19T+67T2 1 + 1.19T + 67T^{2}
71 1+1.26T+71T2 1 + 1.26T + 71T^{2}
73 1−1.73T+73T2 1 - 1.73T + 73T^{2}
79 1+11T+79T2 1 + 11T + 79T^{2}
83 1+10.9T+83T2 1 + 10.9T + 83T^{2}
89 1−8.73T+89T2 1 - 8.73T + 89T^{2}
97 1−5.19T+97T2 1 - 5.19T + 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

−7.67214766810444497897160995480, −7.05273769763900461478915222358, −6.45125998160677866778815476544, −5.74743344142409431839531473547, −4.99461128162831198628129409625, −3.24854940597659777938175090005, −3.01760017158330528165054340899, −1.88714424334233149536197545353, −1.10311351721939928947295414415, 0, 1.10311351721939928947295414415, 1.88714424334233149536197545353, 3.01760017158330528165054340899, 3.24854940597659777938175090005, 4.99461128162831198628129409625, 5.74743344142409431839531473547, 6.45125998160677866778815476544, 7.05273769763900461478915222358, 7.67214766810444497897160995480

Graph of the ZZ-function along the critical line