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Properties

Label 2-920-1.1-c1-0-17
Degree 22
Conductor 920920
Sign 11
Analytic cond. 7.346237.34623
Root an. cond. 2.710392.71039
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.56·3-s + 5-s + 5.12·7-s + 3.56·9-s − 4·11-s − 0.561·13-s + 2.56·15-s − 3.12·17-s + 4·19-s + 13.1·21-s + 23-s + 25-s + 1.43·27-s − 8.56·29-s + 1.43·31-s − 10.2·33-s + 5.12·35-s − 7.12·37-s − 1.43·39-s + 0.561·41-s − 9.12·43-s + 3.56·45-s − 3.68·47-s + 19.2·49-s − 8·51-s − 4.24·53-s − 4·55-s + ⋯
L(s)  = 1  + 1.47·3-s + 0.447·5-s + 1.93·7-s + 1.18·9-s − 1.20·11-s − 0.155·13-s + 0.661·15-s − 0.757·17-s + 0.917·19-s + 2.86·21-s + 0.208·23-s + 0.200·25-s + 0.276·27-s − 1.58·29-s + 0.258·31-s − 1.78·33-s + 0.865·35-s − 1.17·37-s − 0.230·39-s + 0.0876·41-s − 1.39·43-s + 0.530·45-s − 0.537·47-s + 2.74·49-s − 1.12·51-s − 0.583·53-s − 0.539·55-s + ⋯

Functional equation

Λ(s)=(920s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(920s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 920920    =    23⋅5⋅232^{3} \cdot 5 \cdot 23
Sign: 11
Analytic conductor: 7.346237.34623
Root analytic conductor: 2.710392.71039
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 920, ( :1/2), 1)(2,\ 920,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 3.0991317013.099131701
L(12)L(\frac12) ≈\approx 3.0991317013.099131701
L(32)L(\frac{3}{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)
bad2 1 1
5 1−T 1 - T
23 1−T 1 - T
good3 1−2.56T+3T2 1 - 2.56T + 3T^{2}
7 1−5.12T+7T2 1 - 5.12T + 7T^{2}
11 1+4T+11T2 1 + 4T + 11T^{2}
13 1+0.561T+13T2 1 + 0.561T + 13T^{2}
17 1+3.12T+17T2 1 + 3.12T + 17T^{2}
19 1−4T+19T2 1 - 4T + 19T^{2}
29 1+8.56T+29T2 1 + 8.56T + 29T^{2}
31 1−1.43T+31T2 1 - 1.43T + 31T^{2}
37 1+7.12T+37T2 1 + 7.12T + 37T^{2}
41 1−0.561T+41T2 1 - 0.561T + 41T^{2}
43 1+9.12T+43T2 1 + 9.12T + 43T^{2}
47 1+3.68T+47T2 1 + 3.68T + 47T^{2}
53 1+4.24T+53T2 1 + 4.24T + 53T^{2}
59 1+6.24T+59T2 1 + 6.24T + 59T^{2}
61 1−11.1T+61T2 1 - 11.1T + 61T^{2}
67 1−6.24T+67T2 1 - 6.24T + 67T^{2}
71 1−3.68T+71T2 1 - 3.68T + 71T^{2}
73 1−16.5T+73T2 1 - 16.5T + 73T^{2}
79 1+10.2T+79T2 1 + 10.2T + 79T^{2}
83 1−12T+83T2 1 - 12T + 83T^{2}
89 1−10T+89T2 1 - 10T + 89T^{2}
97 1+16.2T+97T2 1 + 16.2T + 97T^{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

−9.920076549679180655210937480297, −9.084272153835367855041594782253, −8.282337019996346633165519184144, −7.86840105021195092401430643316, −7.06434875404388674376261168701, −5.38978448874195467279091161608, −4.84759225581334813187804428545, −3.57529344367571280010444902714, −2.38414527944407314457657815859, −1.71934152560981742499742110002, 1.71934152560981742499742110002, 2.38414527944407314457657815859, 3.57529344367571280010444902714, 4.84759225581334813187804428545, 5.38978448874195467279091161608, 7.06434875404388674376261168701, 7.86840105021195092401430643316, 8.282337019996346633165519184144, 9.084272153835367855041594782253, 9.920076549679180655210937480297

Graph of the ZZ-function along the critical line