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Properties

Label 2-296-1.1-c1-0-3
Degree 22
Conductor 296296
Sign 11
Analytic cond. 2.363572.36357
Root an. cond. 1.537391.53739
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  + 1.46·3-s + 0.462·5-s − 0.323·7-s − 0.860·9-s + 5.58·11-s + 6.58·13-s + 0.676·15-s − 6.64·17-s − 2.64·19-s − 0.473·21-s + 4.86·23-s − 4.78·25-s − 5.64·27-s − 4.86·29-s + 6.46·31-s + 8.16·33-s − 0.149·35-s − 37-s + 9.62·39-s − 0.815·41-s − 1.87·43-s − 0.398·45-s − 1.11·47-s − 6.89·49-s − 9.72·51-s − 12.6·53-s + 2.58·55-s + ⋯
L(s)  = 1  + 0.844·3-s + 0.206·5-s − 0.122·7-s − 0.286·9-s + 1.68·11-s + 1.82·13-s + 0.174·15-s − 1.61·17-s − 0.607·19-s − 0.103·21-s + 1.01·23-s − 0.957·25-s − 1.08·27-s − 0.902·29-s + 1.16·31-s + 1.42·33-s − 0.0252·35-s − 0.164·37-s + 1.54·39-s − 0.127·41-s − 0.285·43-s − 0.0593·45-s − 0.163·47-s − 0.985·49-s − 1.36·51-s − 1.74·53-s + 0.348·55-s + ⋯

Functional equation

Λ(s)=(296s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(296s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 296 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 296296    =    23⋅372^{3} \cdot 37
Sign: 11
Analytic conductor: 2.363572.36357
Root analytic conductor: 1.537391.53739
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 296, ( :1/2), 1)(2,\ 296,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.7551936031.755193603
L(12)L(\frac12) ≈\approx 1.7551936031.755193603
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
37 1+T 1 + T
good3 1−1.46T+3T2 1 - 1.46T + 3T^{2}
5 1−0.462T+5T2 1 - 0.462T + 5T^{2}
7 1+0.323T+7T2 1 + 0.323T + 7T^{2}
11 1−5.58T+11T2 1 - 5.58T + 11T^{2}
13 1−6.58T+13T2 1 - 6.58T + 13T^{2}
17 1+6.64T+17T2 1 + 6.64T + 17T^{2}
19 1+2.64T+19T2 1 + 2.64T + 19T^{2}
23 1−4.86T+23T2 1 - 4.86T + 23T^{2}
29 1+4.86T+29T2 1 + 4.86T + 29T^{2}
31 1−6.46T+31T2 1 - 6.46T + 31T^{2}
41 1+0.815T+41T2 1 + 0.815T + 41T^{2}
43 1+1.87T+43T2 1 + 1.87T + 43T^{2}
47 1+1.11T+47T2 1 + 1.11T + 47T^{2}
53 1+12.6T+53T2 1 + 12.6T + 53T^{2}
59 1+0.128T+59T2 1 + 0.128T + 59T^{2}
61 1+1.10T+61T2 1 + 1.10T + 61T^{2}
67 1−13.4T+67T2 1 - 13.4T + 67T^{2}
71 1+8.04T+71T2 1 + 8.04T + 71T^{2}
73 1−3.58T+73T2 1 - 3.58T + 73T^{2}
79 1+3.78T+79T2 1 + 3.78T + 79T^{2}
83 1+14.6T+83T2 1 + 14.6T + 83T^{2}
89 1−2.14T+89T2 1 - 2.14T + 89T^{2}
97 1−1.05T+97T2 1 - 1.05T + 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

−11.48486625577246254733228686920, −11.08368274153542791459108705379, −9.547408148510392708352804041462, −8.869996196977424984266331586994, −8.293683295342721729660434166900, −6.71769660084259130592038541006, −6.07682723621157549923626312270, −4.27208244246088495069323457097, −3.34243059582778935043951132583, −1.75209515526012064562747615162, 1.75209515526012064562747615162, 3.34243059582778935043951132583, 4.27208244246088495069323457097, 6.07682723621157549923626312270, 6.71769660084259130592038541006, 8.293683295342721729660434166900, 8.869996196977424984266331586994, 9.547408148510392708352804041462, 11.08368274153542791459108705379, 11.48486625577246254733228686920

Graph of the ZZ-function along the critical line