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Properties

Label 2-325-1.1-c5-0-46
Degree 22
Conductor 325325
Sign −1-1
Analytic cond. 52.124752.1247
Root an. cond. 7.219747.21974
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.20·2-s − 29.5·3-s − 21.7·4-s − 94.8·6-s − 11.5·7-s − 172.·8-s + 632.·9-s − 596.·11-s + 642.·12-s + 169·13-s − 37.1·14-s + 142.·16-s + 2.09e3·17-s + 2.02e3·18-s + 35.4·19-s + 342.·21-s − 1.91e3·22-s + 2.78e3·23-s + 5.09e3·24-s + 541.·26-s − 1.15e4·27-s + 251.·28-s − 370.·29-s + 5.05e3·31-s + 5.96e3·32-s + 1.76e4·33-s + 6.71e3·34-s + ⋯
L(s)  = 1  + 0.566·2-s − 1.89·3-s − 0.678·4-s − 1.07·6-s − 0.0892·7-s − 0.951·8-s + 2.60·9-s − 1.48·11-s + 1.28·12-s + 0.277·13-s − 0.0506·14-s + 0.139·16-s + 1.75·17-s + 1.47·18-s + 0.0225·19-s + 0.169·21-s − 0.842·22-s + 1.09·23-s + 1.80·24-s + 0.157·26-s − 3.04·27-s + 0.0605·28-s − 0.0817·29-s + 0.944·31-s + 1.03·32-s + 2.82·33-s + 0.995·34-s + ⋯

Functional equation

Λ(s)=(325s/2 ΓC(s) L(s)=(− Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}
Λ(s)=(325s/2 ΓC(s+5/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 325325    =    52⋅135^{2} \cdot 13
Sign: −1-1
Analytic conductor: 52.124752.1247
Root analytic conductor: 7.219747.21974
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 325, ( :5/2), −1)(2,\ 325,\ (\ :5/2),\ -1)

Particular Values

L(3)L(3) == 00
L(12)L(\frac12) == 00
L(72)L(\frac{7}{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)
bad5 1 1
13 1−169T 1 - 169T
good2 1−3.20T+32T2 1 - 3.20T + 32T^{2}
3 1+29.5T+243T2 1 + 29.5T + 243T^{2}
7 1+11.5T+1.68e4T2 1 + 11.5T + 1.68e4T^{2}
11 1+596.T+1.61e5T2 1 + 596.T + 1.61e5T^{2}
17 1−2.09e3T+1.41e6T2 1 - 2.09e3T + 1.41e6T^{2}
19 1−35.4T+2.47e6T2 1 - 35.4T + 2.47e6T^{2}
23 1−2.78e3T+6.43e6T2 1 - 2.78e3T + 6.43e6T^{2}
29 1+370.T+2.05e7T2 1 + 370.T + 2.05e7T^{2}
31 1−5.05e3T+2.86e7T2 1 - 5.05e3T + 2.86e7T^{2}
37 1−4.12e3T+6.93e7T2 1 - 4.12e3T + 6.93e7T^{2}
41 1+1.81e4T+1.15e8T2 1 + 1.81e4T + 1.15e8T^{2}
43 1+7.90e3T+1.47e8T2 1 + 7.90e3T + 1.47e8T^{2}
47 1+1.31e4T+2.29e8T2 1 + 1.31e4T + 2.29e8T^{2}
53 1−3.82e4T+4.18e8T2 1 - 3.82e4T + 4.18e8T^{2}
59 1−1.78e4T+7.14e8T2 1 - 1.78e4T + 7.14e8T^{2}
61 1−7.39e3T+8.44e8T2 1 - 7.39e3T + 8.44e8T^{2}
67 1+2.33e4T+1.35e9T2 1 + 2.33e4T + 1.35e9T^{2}
71 1+3.31e4T+1.80e9T2 1 + 3.31e4T + 1.80e9T^{2}
73 1+1.08e4T+2.07e9T2 1 + 1.08e4T + 2.07e9T^{2}
79 1+1.77e4T+3.07e9T2 1 + 1.77e4T + 3.07e9T^{2}
83 1+8.42e4T+3.93e9T2 1 + 8.42e4T + 3.93e9T^{2}
89 1+4.64e4T+5.58e9T2 1 + 4.64e4T + 5.58e9T^{2}
97 1−1.54e5T+8.58e9T2 1 - 1.54e5T + 8.58e9T^{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

−10.27662660418272348176641672690, −9.927017616147294011749118418224, −8.285430758176530522835944676595, −7.11302357847818106064733775524, −5.93049367935974027215997573311, −5.30231559823003218437995258480, −4.68655594195722039291424104924, −3.27903928286136950592885603599, −1.03803694548153880708349090320, 0, 1.03803694548153880708349090320, 3.27903928286136950592885603599, 4.68655594195722039291424104924, 5.30231559823003218437995258480, 5.93049367935974027215997573311, 7.11302357847818106064733775524, 8.285430758176530522835944676595, 9.927017616147294011749118418224, 10.27662660418272348176641672690

Graph of the ZZ-function along the critical line