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Properties

Label 2-325-1.1-c5-0-26
Degree 22
Conductor 325325
Sign 11
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 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.16·2-s + 2.56·3-s − 27.2·4-s + 5.57·6-s + 75.5·7-s − 128.·8-s − 236.·9-s + 624.·11-s − 70.1·12-s + 169·13-s + 163.·14-s + 594.·16-s − 2.34e3·17-s − 512.·18-s − 283.·19-s + 194.·21-s + 1.35e3·22-s − 2.04e3·23-s − 330.·24-s + 366.·26-s − 1.23e3·27-s − 2.06e3·28-s + 6.17e3·29-s + 687.·31-s + 5.40e3·32-s + 1.60e3·33-s − 5.08e3·34-s + ⋯
L(s)  = 1  + 0.383·2-s + 0.164·3-s − 0.853·4-s + 0.0631·6-s + 0.583·7-s − 0.710·8-s − 0.972·9-s + 1.55·11-s − 0.140·12-s + 0.277·13-s + 0.223·14-s + 0.580·16-s − 1.96·17-s − 0.372·18-s − 0.180·19-s + 0.0961·21-s + 0.596·22-s − 0.805·23-s − 0.117·24-s + 0.106·26-s − 0.325·27-s − 0.497·28-s + 1.36·29-s + 0.128·31-s + 0.933·32-s + 0.256·33-s − 0.754·34-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: 11
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: 00
Selberg data: (2, 325, ( :5/2), 1)(2,\ 325,\ (\ :5/2),\ 1)

Particular Values

L(3)L(3) ≈\approx 1.9855530741.985553074
L(12)L(\frac12) ≈\approx 1.9855530741.985553074
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 1−2.16T+32T2 1 - 2.16T + 32T^{2}
3 1−2.56T+243T2 1 - 2.56T + 243T^{2}
7 1−75.5T+1.68e4T2 1 - 75.5T + 1.68e4T^{2}
11 1−624.T+1.61e5T2 1 - 624.T + 1.61e5T^{2}
17 1+2.34e3T+1.41e6T2 1 + 2.34e3T + 1.41e6T^{2}
19 1+283.T+2.47e6T2 1 + 283.T + 2.47e6T^{2}
23 1+2.04e3T+6.43e6T2 1 + 2.04e3T + 6.43e6T^{2}
29 1−6.17e3T+2.05e7T2 1 - 6.17e3T + 2.05e7T^{2}
31 1−687.T+2.86e7T2 1 - 687.T + 2.86e7T^{2}
37 1−2.79e3T+6.93e7T2 1 - 2.79e3T + 6.93e7T^{2}
41 1−8.23e3T+1.15e8T2 1 - 8.23e3T + 1.15e8T^{2}
43 1−1.32e4T+1.47e8T2 1 - 1.32e4T + 1.47e8T^{2}
47 1−1.54e4T+2.29e8T2 1 - 1.54e4T + 2.29e8T^{2}
53 1−9.60e3T+4.18e8T2 1 - 9.60e3T + 4.18e8T^{2}
59 1−4.01e4T+7.14e8T2 1 - 4.01e4T + 7.14e8T^{2}
61 1−3.25e4T+8.44e8T2 1 - 3.25e4T + 8.44e8T^{2}
67 1+1.59e4T+1.35e9T2 1 + 1.59e4T + 1.35e9T^{2}
71 1−6.02e4T+1.80e9T2 1 - 6.02e4T + 1.80e9T^{2}
73 1−3.54e4T+2.07e9T2 1 - 3.54e4T + 2.07e9T^{2}
79 1−6.50e4T+3.07e9T2 1 - 6.50e4T + 3.07e9T^{2}
83 1+8.60e4T+3.93e9T2 1 + 8.60e4T + 3.93e9T^{2}
89 1+1.39e5T+5.58e9T2 1 + 1.39e5T + 5.58e9T^{2}
97 1−8.01e3T+8.58e9T2 1 - 8.01e3T + 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

−11.00949111236187945640598256893, −9.603271227361373751662917341526, −8.767665392166807849210134982946, −8.326854363906833199071744918109, −6.68517383539188997965962533736, −5.81703578228648408785844807952, −4.53421273073423733768006889466, −3.87562120534043960472924085283, −2.36928963599832644546840189012, −0.73793836101671324056521906019, 0.73793836101671324056521906019, 2.36928963599832644546840189012, 3.87562120534043960472924085283, 4.53421273073423733768006889466, 5.81703578228648408785844807952, 6.68517383539188997965962533736, 8.326854363906833199071744918109, 8.767665392166807849210134982946, 9.603271227361373751662917341526, 11.00949111236187945640598256893

Graph of the ZZ-function along the critical line