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Properties

Label 2-325-1.1-c5-0-10
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  + 3.78·2-s − 8.20·3-s − 17.6·4-s − 31.0·6-s − 88.9·7-s − 188.·8-s − 175.·9-s − 156.·11-s + 145.·12-s − 169·13-s − 336.·14-s − 145.·16-s − 447.·17-s − 664.·18-s − 269.·19-s + 730.·21-s − 592.·22-s − 1.37e3·23-s + 1.54e3·24-s − 639.·26-s + 3.43e3·27-s + 1.57e3·28-s − 3.69e3·29-s + 797.·31-s + 5.46e3·32-s + 1.28e3·33-s − 1.69e3·34-s + ⋯
L(s)  = 1  + 0.668·2-s − 0.526·3-s − 0.552·4-s − 0.352·6-s − 0.686·7-s − 1.03·8-s − 0.722·9-s − 0.390·11-s + 0.290·12-s − 0.277·13-s − 0.459·14-s − 0.142·16-s − 0.375·17-s − 0.483·18-s − 0.171·19-s + 0.361·21-s − 0.261·22-s − 0.540·23-s + 0.546·24-s − 0.185·26-s + 0.907·27-s + 0.379·28-s − 0.816·29-s + 0.149·31-s + 0.943·32-s + 0.205·33-s − 0.251·34-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 0.75801918010.7580191801
L(12)L(\frac12) ≈\approx 0.75801918010.7580191801
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 1−3.78T+32T2 1 - 3.78T + 32T^{2}
3 1+8.20T+243T2 1 + 8.20T + 243T^{2}
7 1+88.9T+1.68e4T2 1 + 88.9T + 1.68e4T^{2}
11 1+156.T+1.61e5T2 1 + 156.T + 1.61e5T^{2}
17 1+447.T+1.41e6T2 1 + 447.T + 1.41e6T^{2}
19 1+269.T+2.47e6T2 1 + 269.T + 2.47e6T^{2}
23 1+1.37e3T+6.43e6T2 1 + 1.37e3T + 6.43e6T^{2}
29 1+3.69e3T+2.05e7T2 1 + 3.69e3T + 2.05e7T^{2}
31 1−797.T+2.86e7T2 1 - 797.T + 2.86e7T^{2}
37 1−4.39e3T+6.93e7T2 1 - 4.39e3T + 6.93e7T^{2}
41 1−1.43e4T+1.15e8T2 1 - 1.43e4T + 1.15e8T^{2}
43 1+1.13e4T+1.47e8T2 1 + 1.13e4T + 1.47e8T^{2}
47 1−9.97e3T+2.29e8T2 1 - 9.97e3T + 2.29e8T^{2}
53 1+1.15e4T+4.18e8T2 1 + 1.15e4T + 4.18e8T^{2}
59 1−7.13e3T+7.14e8T2 1 - 7.13e3T + 7.14e8T^{2}
61 1−1.66e4T+8.44e8T2 1 - 1.66e4T + 8.44e8T^{2}
67 1+4.21e3T+1.35e9T2 1 + 4.21e3T + 1.35e9T^{2}
71 1−1.28e4T+1.80e9T2 1 - 1.28e4T + 1.80e9T^{2}
73 1−1.30e4T+2.07e9T2 1 - 1.30e4T + 2.07e9T^{2}
79 1−7.47e4T+3.07e9T2 1 - 7.47e4T + 3.07e9T^{2}
83 1+8.54e3T+3.93e9T2 1 + 8.54e3T + 3.93e9T^{2}
89 1−5.95e4T+5.58e9T2 1 - 5.95e4T + 5.58e9T^{2}
97 1−1.73e4T+8.58e9T2 1 - 1.73e4T + 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

−10.93154635370962693217914533373, −9.819359874113385446827075130203, −9.006768629930498181630075410956, −7.947641408221659107676034296682, −6.50518251108565296809626075942, −5.75187768145654263500815590950, −4.85812672178506052188292407225, −3.70912869418540935032313904278, −2.58187854116373633571287599977, −0.42496638566140978183476951573, 0.42496638566140978183476951573, 2.58187854116373633571287599977, 3.70912869418540935032313904278, 4.85812672178506052188292407225, 5.75187768145654263500815590950, 6.50518251108565296809626075942, 7.947641408221659107676034296682, 9.006768629930498181630075410956, 9.819359874113385446827075130203, 10.93154635370962693217914533373

Graph of the ZZ-function along the critical line