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Properties

Label 2-4788-1.1-c1-0-15
Degree 22
Conductor 47884788
Sign 11
Analytic cond. 38.232338.2323
Root an. cond. 6.183236.18323
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  + 1.79·5-s − 7-s + 3.83·11-s − 3.43·13-s + 0.293·17-s − 19-s + 6.83·23-s − 1.78·25-s + 5.62·29-s − 4.78·31-s − 1.79·35-s + 11.5·37-s − 5.57·41-s + 4.78·43-s − 12.7·47-s + 49-s + 4.12·53-s + 6.87·55-s + 11.5·61-s − 6.16·65-s + 10.2·67-s + 3.53·71-s − 10.4·73-s − 3.83·77-s − 2.22·79-s + 17.7·83-s + 0.526·85-s + ⋯
L(s)  = 1  + 0.802·5-s − 0.377·7-s + 1.15·11-s − 0.953·13-s + 0.0711·17-s − 0.229·19-s + 1.42·23-s − 0.356·25-s + 1.04·29-s − 0.859·31-s − 0.303·35-s + 1.90·37-s − 0.871·41-s + 0.729·43-s − 1.85·47-s + 0.142·49-s + 0.566·53-s + 0.927·55-s + 1.48·61-s − 0.764·65-s + 1.24·67-s + 0.420·71-s − 1.22·73-s − 0.436·77-s − 0.249·79-s + 1.95·83-s + 0.0571·85-s + ⋯

Functional equation

Λ(s)=(4788s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(4788s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 47884788    =    22⋅32⋅7⋅192^{2} \cdot 3^{2} \cdot 7 \cdot 19
Sign: 11
Analytic conductor: 38.232338.2323
Root analytic conductor: 6.183236.18323
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 4788, ( :1/2), 1)(2,\ 4788,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.2946728312.294672831
L(12)L(\frac12) ≈\approx 2.2946728312.294672831
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
3 1 1
7 1+T 1 + T
19 1+T 1 + T
good5 1−1.79T+5T2 1 - 1.79T + 5T^{2}
11 1−3.83T+11T2 1 - 3.83T + 11T^{2}
13 1+3.43T+13T2 1 + 3.43T + 13T^{2}
17 1−0.293T+17T2 1 - 0.293T + 17T^{2}
23 1−6.83T+23T2 1 - 6.83T + 23T^{2}
29 1−5.62T+29T2 1 - 5.62T + 29T^{2}
31 1+4.78T+31T2 1 + 4.78T + 31T^{2}
37 1−11.5T+37T2 1 - 11.5T + 37T^{2}
41 1+5.57T+41T2 1 + 5.57T + 41T^{2}
43 1−4.78T+43T2 1 - 4.78T + 43T^{2}
47 1+12.7T+47T2 1 + 12.7T + 47T^{2}
53 1−4.12T+53T2 1 - 4.12T + 53T^{2}
59 1+59T2 1 + 59T^{2}
61 1−11.5T+61T2 1 - 11.5T + 61T^{2}
67 1−10.2T+67T2 1 - 10.2T + 67T^{2}
71 1−3.53T+71T2 1 - 3.53T + 71T^{2}
73 1+10.4T+73T2 1 + 10.4T + 73T^{2}
79 1+2.22T+79T2 1 + 2.22T + 79T^{2}
83 1−17.7T+83T2 1 - 17.7T + 83T^{2}
89 1−5.08T+89T2 1 - 5.08T + 89T^{2}
97 1−9.52T+97T2 1 - 9.52T + 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

−8.382070942984535254732356641169, −7.43705186614841804902617808809, −6.71386643370478455615051921590, −6.25131841746072517554080176928, −5.35247709401873527083703035981, −4.65645194978473531664161392639, −3.72525572088040779686104705666, −2.80320590836861123465855947910, −1.95305875134974003450157961823, −0.852901522617359620717966458676, 0.852901522617359620717966458676, 1.95305875134974003450157961823, 2.80320590836861123465855947910, 3.72525572088040779686104705666, 4.65645194978473531664161392639, 5.35247709401873527083703035981, 6.25131841746072517554080176928, 6.71386643370478455615051921590, 7.43705186614841804902617808809, 8.382070942984535254732356641169

Graph of the ZZ-function along the critical line