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Properties

Label 2-325-1.1-c5-0-91
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  + 8.23·2-s + 11.6·3-s + 35.8·4-s + 95.8·6-s − 195.·7-s + 31.9·8-s − 107.·9-s + 64.5·11-s + 417.·12-s + 169·13-s − 1.60e3·14-s − 885.·16-s + 426.·17-s − 886.·18-s − 959.·19-s − 2.27e3·21-s + 531.·22-s − 499.·23-s + 371.·24-s + 1.39e3·26-s − 4.07e3·27-s − 6.99e3·28-s + 1.28e3·29-s − 6.73e3·31-s − 8.31e3·32-s + 751.·33-s + 3.50e3·34-s + ⋯
L(s)  = 1  + 1.45·2-s + 0.746·3-s + 1.12·4-s + 1.08·6-s − 1.50·7-s + 0.176·8-s − 0.442·9-s + 0.160·11-s + 0.836·12-s + 0.277·13-s − 2.19·14-s − 0.864·16-s + 0.357·17-s − 0.644·18-s − 0.609·19-s − 1.12·21-s + 0.234·22-s − 0.196·23-s + 0.131·24-s + 0.403·26-s − 1.07·27-s − 1.68·28-s + 0.284·29-s − 1.25·31-s − 1.43·32-s + 0.120·33-s + 0.520·34-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 1−8.23T+32T2 1 - 8.23T + 32T^{2}
3 1−11.6T+243T2 1 - 11.6T + 243T^{2}
7 1+195.T+1.68e4T2 1 + 195.T + 1.68e4T^{2}
11 1−64.5T+1.61e5T2 1 - 64.5T + 1.61e5T^{2}
17 1−426.T+1.41e6T2 1 - 426.T + 1.41e6T^{2}
19 1+959.T+2.47e6T2 1 + 959.T + 2.47e6T^{2}
23 1+499.T+6.43e6T2 1 + 499.T + 6.43e6T^{2}
29 1−1.28e3T+2.05e7T2 1 - 1.28e3T + 2.05e7T^{2}
31 1+6.73e3T+2.86e7T2 1 + 6.73e3T + 2.86e7T^{2}
37 1+6.21e3T+6.93e7T2 1 + 6.21e3T + 6.93e7T^{2}
41 1+6.49e3T+1.15e8T2 1 + 6.49e3T + 1.15e8T^{2}
43 1+1.56e4T+1.47e8T2 1 + 1.56e4T + 1.47e8T^{2}
47 1+6.29e3T+2.29e8T2 1 + 6.29e3T + 2.29e8T^{2}
53 1−4.03e4T+4.18e8T2 1 - 4.03e4T + 4.18e8T^{2}
59 1−2.56e4T+7.14e8T2 1 - 2.56e4T + 7.14e8T^{2}
61 1−2.41e4T+8.44e8T2 1 - 2.41e4T + 8.44e8T^{2}
67 1+3.91e4T+1.35e9T2 1 + 3.91e4T + 1.35e9T^{2}
71 1+3.26e4T+1.80e9T2 1 + 3.26e4T + 1.80e9T^{2}
73 1−1.45e4T+2.07e9T2 1 - 1.45e4T + 2.07e9T^{2}
79 1−7.90e4T+3.07e9T2 1 - 7.90e4T + 3.07e9T^{2}
83 1−1.02e5T+3.93e9T2 1 - 1.02e5T + 3.93e9T^{2}
89 1+4.81e4T+5.58e9T2 1 + 4.81e4T + 5.58e9T^{2}
97 1−7.33e4T+8.58e9T2 1 - 7.33e4T + 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

−10.36135116299176545805545148329, −9.303601173925032901600279594453, −8.511630872512766324982040592189, −7.02750572149212878156652535519, −6.21090166456278466689259959614, −5.30949018839157577329638233947, −3.79661444868374504347434088296, −3.32915391293092424549672969131, −2.27009398876332768304454620923, 0, 2.27009398876332768304454620923, 3.32915391293092424549672969131, 3.79661444868374504347434088296, 5.30949018839157577329638233947, 6.21090166456278466689259959614, 7.02750572149212878156652535519, 8.511630872512766324982040592189, 9.303601173925032901600279594453, 10.36135116299176545805545148329

Graph of the ZZ-function along the critical line