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Properties

Label 2-325-1.1-c5-0-87
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  + 4.30·2-s + 19.9·3-s − 13.4·4-s + 85.7·6-s + 125.·7-s − 195.·8-s + 153.·9-s − 709.·11-s − 267.·12-s + 169·13-s + 541.·14-s − 412.·16-s − 1.92e3·17-s + 660.·18-s − 2.16e3·19-s + 2.50e3·21-s − 3.05e3·22-s − 305.·23-s − 3.89e3·24-s + 727.·26-s − 1.78e3·27-s − 1.69e3·28-s + 4.51e3·29-s + 4.07e3·31-s + 4.48e3·32-s − 1.41e4·33-s − 8.29e3·34-s + ⋯
L(s)  = 1  + 0.761·2-s + 1.27·3-s − 0.420·4-s + 0.972·6-s + 0.970·7-s − 1.08·8-s + 0.631·9-s − 1.76·11-s − 0.537·12-s + 0.277·13-s + 0.738·14-s − 0.402·16-s − 1.61·17-s + 0.480·18-s − 1.37·19-s + 1.23·21-s − 1.34·22-s − 0.120·23-s − 1.38·24-s + 0.211·26-s − 0.471·27-s − 0.408·28-s + 0.996·29-s + 0.762·31-s + 0.774·32-s − 2.25·33-s − 1.23·34-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 1−4.30T+32T2 1 - 4.30T + 32T^{2}
3 1−19.9T+243T2 1 - 19.9T + 243T^{2}
7 1−125.T+1.68e4T2 1 - 125.T + 1.68e4T^{2}
11 1+709.T+1.61e5T2 1 + 709.T + 1.61e5T^{2}
17 1+1.92e3T+1.41e6T2 1 + 1.92e3T + 1.41e6T^{2}
19 1+2.16e3T+2.47e6T2 1 + 2.16e3T + 2.47e6T^{2}
23 1+305.T+6.43e6T2 1 + 305.T + 6.43e6T^{2}
29 1−4.51e3T+2.05e7T2 1 - 4.51e3T + 2.05e7T^{2}
31 1−4.07e3T+2.86e7T2 1 - 4.07e3T + 2.86e7T^{2}
37 1+1.23e4T+6.93e7T2 1 + 1.23e4T + 6.93e7T^{2}
41 1−1.44e4T+1.15e8T2 1 - 1.44e4T + 1.15e8T^{2}
43 1−1.71e4T+1.47e8T2 1 - 1.71e4T + 1.47e8T^{2}
47 1+1.83e4T+2.29e8T2 1 + 1.83e4T + 2.29e8T^{2}
53 1+1.19e4T+4.18e8T2 1 + 1.19e4T + 4.18e8T^{2}
59 1+3.98e4T+7.14e8T2 1 + 3.98e4T + 7.14e8T^{2}
61 1−4.49e3T+8.44e8T2 1 - 4.49e3T + 8.44e8T^{2}
67 1+3.04e4T+1.35e9T2 1 + 3.04e4T + 1.35e9T^{2}
71 1+1.45e4T+1.80e9T2 1 + 1.45e4T + 1.80e9T^{2}
73 1−5.90e4T+2.07e9T2 1 - 5.90e4T + 2.07e9T^{2}
79 1+4.33e4T+3.07e9T2 1 + 4.33e4T + 3.07e9T^{2}
83 1+8.38e4T+3.93e9T2 1 + 8.38e4T + 3.93e9T^{2}
89 1−1.24e5T+5.58e9T2 1 - 1.24e5T + 5.58e9T^{2}
97 1−9.38e4T+8.58e9T2 1 - 9.38e4T + 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

−10.35699232180373174201953123910, −9.009129930207136535761959963943, −8.453739483802988699528983485175, −7.77605092608974429902220953248, −6.23015161783151238353249644833, −4.91611368465024590704238974851, −4.28006057080907792511649350576, −2.92274467689949341862851432910, −2.13332138496309952487703253777, 0, 2.13332138496309952487703253777, 2.92274467689949341862851432910, 4.28006057080907792511649350576, 4.91611368465024590704238974851, 6.23015161783151238353249644833, 7.77605092608974429902220953248, 8.453739483802988699528983485175, 9.009129930207136535761959963943, 10.35699232180373174201953123910

Graph of the ZZ-function along the critical line