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Properties

Label 2-4788-1.1-c1-0-4
Degree 22
Conductor 47884788
Sign 11
Analytic cond. 38.232338.2323
Root an. cond. 6.183236.18323
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s − 7-s − 6·11-s + 2·13-s + 2·17-s + 19-s + 2·23-s − 25-s − 4·29-s + 2·35-s − 2·37-s − 8·43-s − 6·47-s + 49-s + 4·53-s + 12·55-s + 2·61-s − 4·65-s + 4·67-s − 12·71-s + 6·73-s + 6·77-s + 8·79-s − 10·83-s − 4·85-s + 8·89-s − 2·91-s + ⋯
L(s)  = 1  − 0.894·5-s − 0.377·7-s − 1.80·11-s + 0.554·13-s + 0.485·17-s + 0.229·19-s + 0.417·23-s − 1/5·25-s − 0.742·29-s + 0.338·35-s − 0.328·37-s − 1.21·43-s − 0.875·47-s + 1/7·49-s + 0.549·53-s + 1.61·55-s + 0.256·61-s − 0.496·65-s + 0.488·67-s − 1.42·71-s + 0.702·73-s + 0.683·77-s + 0.900·79-s − 1.09·83-s − 0.433·85-s + 0.847·89-s − 0.209·91-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: yes
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 0.91690303830.9169030383
L(12)L(\frac12) ≈\approx 0.91690303830.9169030383
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+2T+pT2 1 + 2 T + p T^{2}
11 1+6T+pT2 1 + 6 T + p T^{2}
13 1−2T+pT2 1 - 2 T + p T^{2}
17 1−2T+pT2 1 - 2 T + p T^{2}
23 1−2T+pT2 1 - 2 T + p T^{2}
29 1+4T+pT2 1 + 4 T + p T^{2}
31 1+pT2 1 + p T^{2}
37 1+2T+pT2 1 + 2 T + p T^{2}
41 1+pT2 1 + p T^{2}
43 1+8T+pT2 1 + 8 T + p T^{2}
47 1+6T+pT2 1 + 6 T + p T^{2}
53 1−4T+pT2 1 - 4 T + p T^{2}
59 1+pT2 1 + p T^{2}
61 1−2T+pT2 1 - 2 T + p T^{2}
67 1−4T+pT2 1 - 4 T + p T^{2}
71 1+12T+pT2 1 + 12 T + p T^{2}
73 1−6T+pT2 1 - 6 T + p T^{2}
79 1−8T+pT2 1 - 8 T + p T^{2}
83 1+10T+pT2 1 + 10 T + p T^{2}
89 1−8T+pT2 1 - 8 T + p T^{2}
97 1+2T+pT2 1 + 2 T + p T^{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

−8.184639677331554303933704650434, −7.61588901075909735518418162462, −7.04778130859601909544589177316, −6.02260122758311729025741163435, −5.33160831109524633136613980263, −4.62328709237547491330504325182, −3.56014806445052128856172122648, −3.10537312677644226539074053260, −1.98071470066691974083222620926, −0.50684573635306972930557790284, 0.50684573635306972930557790284, 1.98071470066691974083222620926, 3.10537312677644226539074053260, 3.56014806445052128856172122648, 4.62328709237547491330504325182, 5.33160831109524633136613980263, 6.02260122758311729025741163435, 7.04778130859601909544589177316, 7.61588901075909735518418162462, 8.184639677331554303933704650434

Graph of the ZZ-function along the critical line