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Properties

Label 2-1881-1.1-c1-0-19
Degree 22
Conductor 18811881
Sign 11
Analytic cond. 15.019815.0198
Root an. cond. 3.875543.87554
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  + 1.19·2-s − 0.576·4-s − 1.08·5-s − 1.30·7-s − 3.07·8-s − 1.29·10-s + 11-s + 2.53·13-s − 1.56·14-s − 2.51·16-s + 5.43·17-s + 19-s + 0.623·20-s + 1.19·22-s − 3.87·23-s − 3.82·25-s + 3.03·26-s + 0.754·28-s + 2.41·29-s + 3.03·31-s + 3.14·32-s + 6.48·34-s + 1.41·35-s + 6.85·37-s + 1.19·38-s + 3.32·40-s + 6.11·41-s + ⋯
L(s)  = 1  + 0.843·2-s − 0.288·4-s − 0.484·5-s − 0.494·7-s − 1.08·8-s − 0.408·10-s + 0.301·11-s + 0.704·13-s − 0.417·14-s − 0.628·16-s + 1.31·17-s + 0.229·19-s + 0.139·20-s + 0.254·22-s − 0.807·23-s − 0.765·25-s + 0.594·26-s + 0.142·28-s + 0.448·29-s + 0.545·31-s + 0.556·32-s + 1.11·34-s + 0.239·35-s + 1.12·37-s + 0.193·38-s + 0.526·40-s + 0.954·41-s + ⋯

Functional equation

Λ(s)=(1881s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1881 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1881s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1881 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 18811881    =    32⋅11⋅193^{2} \cdot 11 \cdot 19
Sign: 11
Analytic conductor: 15.019815.0198
Root analytic conductor: 3.875543.87554
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1881, ( :1/2), 1)(2,\ 1881,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.9187784821.918778482
L(12)L(\frac12) ≈\approx 1.9187784821.918778482
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)
bad3 1 1
11 1−T 1 - T
19 1−T 1 - T
good2 1−1.19T+2T2 1 - 1.19T + 2T^{2}
5 1+1.08T+5T2 1 + 1.08T + 5T^{2}
7 1+1.30T+7T2 1 + 1.30T + 7T^{2}
13 1−2.53T+13T2 1 - 2.53T + 13T^{2}
17 1−5.43T+17T2 1 - 5.43T + 17T^{2}
23 1+3.87T+23T2 1 + 3.87T + 23T^{2}
29 1−2.41T+29T2 1 - 2.41T + 29T^{2}
31 1−3.03T+31T2 1 - 3.03T + 31T^{2}
37 1−6.85T+37T2 1 - 6.85T + 37T^{2}
41 1−6.11T+41T2 1 - 6.11T + 41T^{2}
43 1+2.95T+43T2 1 + 2.95T + 43T^{2}
47 1−12.0T+47T2 1 - 12.0T + 47T^{2}
53 1−0.992T+53T2 1 - 0.992T + 53T^{2}
59 1−14.2T+59T2 1 - 14.2T + 59T^{2}
61 1+5.82T+61T2 1 + 5.82T + 61T^{2}
67 1−8.79T+67T2 1 - 8.79T + 67T^{2}
71 1−2.44T+71T2 1 - 2.44T + 71T^{2}
73 1−5.84T+73T2 1 - 5.84T + 73T^{2}
79 1−17.0T+79T2 1 - 17.0T + 79T^{2}
83 1−7.01T+83T2 1 - 7.01T + 83T^{2}
89 1−9.13T+89T2 1 - 9.13T + 89T^{2}
97 1+14.7T+97T2 1 + 14.7T + 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

−9.367873363522709536639391842119, −8.306008344928859887793128379878, −7.77425248603725662767184238162, −6.56274010481225011859369230947, −5.93020880892929800496132444305, −5.17723017347159556447726811935, −4.01553981921566074711944945812, −3.69749194964756242402871313388, −2.62949448611511245027306210203, −0.837710298294896467161867034901, 0.837710298294896467161867034901, 2.62949448611511245027306210203, 3.69749194964756242402871313388, 4.01553981921566074711944945812, 5.17723017347159556447726811935, 5.93020880892929800496132444305, 6.56274010481225011859369230947, 7.77425248603725662767184238162, 8.306008344928859887793128379878, 9.367873363522709536639391842119

Graph of the ZZ-function along the critical line