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Properties

Label 2-28e2-1.1-c5-0-56
Degree 22
Conductor 784784
Sign 11
Analytic cond. 125.740125.740
Root an. cond. 11.213411.2134
Motivic weight 55
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 20·3-s + 74·5-s + 157·9-s − 124·11-s − 478·13-s + 1.48e3·15-s + 1.19e3·17-s + 3.04e3·19-s − 184·23-s + 2.35e3·25-s − 1.72e3·27-s − 3.28e3·29-s − 5.72e3·31-s − 2.48e3·33-s + 1.03e4·37-s − 9.56e3·39-s + 8.88e3·41-s + 9.18e3·43-s + 1.16e4·45-s + 2.36e4·47-s + 2.39e4·51-s + 1.16e4·53-s − 9.17e3·55-s + 6.08e4·57-s + 1.68e4·59-s + 1.84e4·61-s − 3.53e4·65-s + ⋯
L(s)  = 1  + 1.28·3-s + 1.32·5-s + 0.646·9-s − 0.308·11-s − 0.784·13-s + 1.69·15-s + 1.00·17-s + 1.93·19-s − 0.0725·23-s + 0.752·25-s − 0.454·27-s − 0.724·29-s − 1.07·31-s − 0.396·33-s + 1.24·37-s − 1.00·39-s + 0.825·41-s + 0.757·43-s + 0.855·45-s + 1.56·47-s + 1.28·51-s + 0.571·53-s − 0.409·55-s + 2.48·57-s + 0.631·59-s + 0.635·61-s − 1.03·65-s + ⋯

Functional equation

Λ(s)=(784s/2 ΓC(s) L(s)=( Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(784s/2 ΓC(s+5/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 784784    =    24⋅722^{4} \cdot 7^{2}
Sign: 11
Analytic conductor: 125.740125.740
Root analytic conductor: 11.213411.2134
Motivic weight: 55
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 784, ( :5/2), 1)(2,\ 784,\ (\ :5/2),\ 1)

Particular Values

L(3)L(3) ≈\approx 5.0910050475.091005047
L(12)L(\frac12) ≈\approx 5.0910050475.091005047
L(72)L(\frac{7}{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
7 1 1
good3 1−20T+p5T2 1 - 20 T + p^{5} T^{2}
5 1−74T+p5T2 1 - 74 T + p^{5} T^{2}
11 1+124T+p5T2 1 + 124 T + p^{5} T^{2}
13 1+478T+p5T2 1 + 478 T + p^{5} T^{2}
17 1−1198T+p5T2 1 - 1198 T + p^{5} T^{2}
19 1−3044T+p5T2 1 - 3044 T + p^{5} T^{2}
23 1+8pT+p5T2 1 + 8 p T + p^{5} T^{2}
29 1+3282T+p5T2 1 + 3282 T + p^{5} T^{2}
31 1+5728T+p5T2 1 + 5728 T + p^{5} T^{2}
37 1−10326T+p5T2 1 - 10326 T + p^{5} T^{2}
41 1−8886T+p5T2 1 - 8886 T + p^{5} T^{2}
43 1−9188T+p5T2 1 - 9188 T + p^{5} T^{2}
47 1−23664T+p5T2 1 - 23664 T + p^{5} T^{2}
53 1−11686T+p5T2 1 - 11686 T + p^{5} T^{2}
59 1−16876T+p5T2 1 - 16876 T + p^{5} T^{2}
61 1−18482T+p5T2 1 - 18482 T + p^{5} T^{2}
67 1−15532T+p5T2 1 - 15532 T + p^{5} T^{2}
71 1−31960T+p5T2 1 - 31960 T + p^{5} T^{2}
73 1−4886T+p5T2 1 - 4886 T + p^{5} T^{2}
79 1+44560T+p5T2 1 + 44560 T + p^{5} T^{2}
83 1−67364T+p5T2 1 - 67364 T + p^{5} T^{2}
89 1+71994T+p5T2 1 + 71994 T + p^{5} T^{2}
97 1+48866T+p5T2 1 + 48866 T + p^{5} 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

−9.684293237209611819458237506435, −8.898451979725293816946711617242, −7.71658479618322588105628688743, −7.32126689258511098639847932970, −5.78772713605093594021794091856, −5.31998942044890269830265833872, −3.81029654867682447348185824664, −2.79556429477722180003451147058, −2.16935983469153455750546168408, −1.01074162202961239920134415201, 1.01074162202961239920134415201, 2.16935983469153455750546168408, 2.79556429477722180003451147058, 3.81029654867682447348185824664, 5.31998942044890269830265833872, 5.78772713605093594021794091856, 7.32126689258511098639847932970, 7.71658479618322588105628688743, 8.898451979725293816946711617242, 9.684293237209611819458237506435

Graph of the ZZ-function along the critical line