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Properties

Label 2-4788-1.1-c1-0-25
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  + 3.82·5-s + 7-s + 5.86·11-s − 0.605·13-s − 2.04·17-s + 19-s + 5.86·23-s + 9.60·25-s − 2.04·29-s − 6.60·31-s + 3.82·35-s + 2·37-s + 5.86·41-s − 6.60·43-s + 2.04·47-s + 49-s + 3.82·53-s + 22.4·55-s + 11.7·59-s + 2·61-s − 2.31·65-s − 9.21·67-s − 15.5·71-s + 7.21·73-s + 5.86·77-s − 4·79-s − 3.82·83-s + ⋯
L(s)  = 1  + 1.70·5-s + 0.377·7-s + 1.76·11-s − 0.167·13-s − 0.496·17-s + 0.229·19-s + 1.22·23-s + 1.92·25-s − 0.379·29-s − 1.18·31-s + 0.645·35-s + 0.328·37-s + 0.916·41-s − 1.00·43-s + 0.298·47-s + 0.142·49-s + 0.524·53-s + 3.02·55-s + 1.52·59-s + 0.256·61-s − 0.287·65-s − 1.12·67-s − 1.84·71-s + 0.843·73-s + 0.668·77-s − 0.450·79-s − 0.419·83-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 3.4098256613.409825661
L(12)L(\frac12) ≈\approx 3.4098256613.409825661
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 1−3.82T+5T2 1 - 3.82T + 5T^{2}
11 1−5.86T+11T2 1 - 5.86T + 11T^{2}
13 1+0.605T+13T2 1 + 0.605T + 13T^{2}
17 1+2.04T+17T2 1 + 2.04T + 17T^{2}
23 1−5.86T+23T2 1 - 5.86T + 23T^{2}
29 1+2.04T+29T2 1 + 2.04T + 29T^{2}
31 1+6.60T+31T2 1 + 6.60T + 31T^{2}
37 1−2T+37T2 1 - 2T + 37T^{2}
41 1−5.86T+41T2 1 - 5.86T + 41T^{2}
43 1+6.60T+43T2 1 + 6.60T + 43T^{2}
47 1−2.04T+47T2 1 - 2.04T + 47T^{2}
53 1−3.82T+53T2 1 - 3.82T + 53T^{2}
59 1−11.7T+59T2 1 - 11.7T + 59T^{2}
61 1−2T+61T2 1 - 2T + 61T^{2}
67 1+9.21T+67T2 1 + 9.21T + 67T^{2}
71 1+15.5T+71T2 1 + 15.5T + 71T^{2}
73 1−7.21T+73T2 1 - 7.21T + 73T^{2}
79 1+4T+79T2 1 + 4T + 79T^{2}
83 1+3.82T+83T2 1 + 3.82T + 83T^{2}
89 1+9.41T+89T2 1 + 9.41T + 89T^{2}
97 1+3.21T+97T2 1 + 3.21T + 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

−8.635022662518162387235942433003, −7.28410495483941561353502992561, −6.82128666934031340362514529596, −6.05422123082313123851063337317, −5.49020893736721232574616467332, −4.67589074170689324771630098510, −3.78260056896974884122769912922, −2.69730918150489330677421750664, −1.79845129414368275112248748901, −1.14812882339191650853659708606, 1.14812882339191650853659708606, 1.79845129414368275112248748901, 2.69730918150489330677421750664, 3.78260056896974884122769912922, 4.67589074170689324771630098510, 5.49020893736721232574616467332, 6.05422123082313123851063337317, 6.82128666934031340362514529596, 7.28410495483941561353502992561, 8.635022662518162387235942433003

Graph of the ZZ-function along the critical line