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Properties

Label 2-28e2-1.1-c5-0-52
Degree 22
Conductor 784784
Sign −1-1
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 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 16·3-s − 16·5-s + 13·9-s + 76·11-s − 880·13-s + 256·15-s + 1.05e3·17-s + 1.93e3·19-s − 936·23-s − 2.86e3·25-s + 3.68e3·27-s − 3.98e3·29-s + 1.56e3·31-s − 1.21e3·33-s + 4.93e3·37-s + 1.40e4·39-s + 1.58e4·41-s + 1.64e4·43-s − 208·45-s − 2.07e4·47-s − 1.68e4·51-s − 3.74e4·53-s − 1.21e3·55-s − 3.09e4·57-s + 2.11e4·59-s + 2.99e3·61-s + 1.40e4·65-s + ⋯
L(s)  = 1  − 1.02·3-s − 0.286·5-s + 0.0534·9-s + 0.189·11-s − 1.44·13-s + 0.293·15-s + 0.886·17-s + 1.23·19-s − 0.368·23-s − 0.918·25-s + 0.971·27-s − 0.879·29-s + 0.293·31-s − 0.194·33-s + 0.592·37-s + 1.48·39-s + 1.47·41-s + 1.35·43-s − 0.0153·45-s − 1.37·47-s − 0.909·51-s − 1.82·53-s − 0.0542·55-s − 1.26·57-s + 0.790·59-s + 0.102·61-s + 0.413·65-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: −1-1
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: 11
Selberg data: (2, 784, ( :5/2), −1)(2,\ 784,\ (\ :5/2),\ -1)

Particular Values

L(3)L(3) == 00
L(12)L(\frac12) == 00
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+16T+p5T2 1 + 16 T + p^{5} T^{2}
5 1+16T+p5T2 1 + 16 T + p^{5} T^{2}
11 1−76T+p5T2 1 - 76 T + p^{5} T^{2}
13 1+880T+p5T2 1 + 880 T + p^{5} T^{2}
17 1−1056T+p5T2 1 - 1056 T + p^{5} T^{2}
19 1−1936T+p5T2 1 - 1936 T + p^{5} T^{2}
23 1+936T+p5T2 1 + 936 T + p^{5} T^{2}
29 1+3982T+p5T2 1 + 3982 T + p^{5} T^{2}
31 1−1568T+p5T2 1 - 1568 T + p^{5} T^{2}
37 1−4938T+p5T2 1 - 4938 T + p^{5} T^{2}
41 1−15840T+p5T2 1 - 15840 T + p^{5} T^{2}
43 1−16412T+p5T2 1 - 16412 T + p^{5} T^{2}
47 1+20768T+p5T2 1 + 20768 T + p^{5} T^{2}
53 1+37402T+p5T2 1 + 37402 T + p^{5} T^{2}
59 1−21136T+p5T2 1 - 21136 T + p^{5} T^{2}
61 1−2992T+p5T2 1 - 2992 T + p^{5} T^{2}
67 1−45836T+p5T2 1 - 45836 T + p^{5} T^{2}
71 1−49840T+p5T2 1 - 49840 T + p^{5} T^{2}
73 1−56320T+p5T2 1 - 56320 T + p^{5} T^{2}
79 1+40744T+p5T2 1 + 40744 T + p^{5} T^{2}
83 1−112464T+p5T2 1 - 112464 T + p^{5} T^{2}
89 1+64256T+p5T2 1 + 64256 T + p^{5} T^{2}
97 1−2272T+p5T2 1 - 2272 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

−9.480734054617966802332604679379, −7.957943204506859943187598669981, −7.46499361889830171718590638376, −6.34731097201404301510152403681, −5.51394436598893893263245732234, −4.84574670329281300904812797432, −3.65705436260426689468554717478, −2.45183153771855337641416495382, −0.975926757781164449322748518944, 0, 0.975926757781164449322748518944, 2.45183153771855337641416495382, 3.65705436260426689468554717478, 4.84574670329281300904812797432, 5.51394436598893893263245732234, 6.34731097201404301510152403681, 7.46499361889830171718590638376, 7.957943204506859943187598669981, 9.480734054617966802332604679379

Graph of the ZZ-function along the critical line