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Properties

Label 2-56-1.1-c5-0-4
Degree 22
Conductor 5656
Sign 11
Analytic cond. 8.981498.98149
Root an. cond. 2.996912.99691
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  + 30·3-s + 32·5-s + 49·7-s + 657·9-s − 624·11-s − 708·13-s + 960·15-s + 934·17-s + 1.85e3·19-s + 1.47e3·21-s − 1.12e3·23-s − 2.10e3·25-s + 1.24e4·27-s − 1.17e3·29-s + 2.90e3·31-s − 1.87e4·33-s + 1.56e3·35-s − 1.24e4·37-s − 2.12e4·39-s + 2.66e3·41-s − 7.14e3·43-s + 2.10e4·45-s − 7.46e3·47-s + 2.40e3·49-s + 2.80e4·51-s − 2.72e4·53-s − 1.99e4·55-s + ⋯
L(s)  = 1  + 1.92·3-s + 0.572·5-s + 0.377·7-s + 2.70·9-s − 1.55·11-s − 1.16·13-s + 1.10·15-s + 0.783·17-s + 1.18·19-s + 0.727·21-s − 0.441·23-s − 0.672·25-s + 3.27·27-s − 0.259·29-s + 0.543·31-s − 2.99·33-s + 0.216·35-s − 1.49·37-s − 2.23·39-s + 0.247·41-s − 0.589·43-s + 1.54·45-s − 0.493·47-s + 1/7·49-s + 1.50·51-s − 1.33·53-s − 0.890·55-s + ⋯

Functional equation

Λ(s)=(56s/2 ΓC(s) L(s)=( Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(56s/2 ΓC(s+5/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5656    =    23⋅72^{3} \cdot 7
Sign: 11
Analytic conductor: 8.981498.98149
Root analytic conductor: 2.996912.99691
Motivic weight: 55
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 56, ( :5/2), 1)(2,\ 56,\ (\ :5/2),\ 1)

Particular Values

L(3)L(3) ≈\approx 3.2477620683.247762068
L(12)L(\frac12) ≈\approx 3.2477620683.247762068
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−p2T 1 - p^{2} T
good3 1−10pT+p5T2 1 - 10 p T + p^{5} T^{2}
5 1−32T+p5T2 1 - 32 T + p^{5} T^{2}
11 1+624T+p5T2 1 + 624 T + p^{5} T^{2}
13 1+708T+p5T2 1 + 708 T + p^{5} T^{2}
17 1−934T+p5T2 1 - 934 T + p^{5} T^{2}
19 1−1858T+p5T2 1 - 1858 T + p^{5} T^{2}
23 1+1120T+p5T2 1 + 1120 T + p^{5} T^{2}
29 1+1174T+p5T2 1 + 1174 T + p^{5} T^{2}
31 1−2908T+p5T2 1 - 2908 T + p^{5} T^{2}
37 1+12462T+p5T2 1 + 12462 T + p^{5} T^{2}
41 1−2662T+p5T2 1 - 2662 T + p^{5} T^{2}
43 1+7144T+p5T2 1 + 7144 T + p^{5} T^{2}
47 1+7468T+p5T2 1 + 7468 T + p^{5} T^{2}
53 1+27274T+p5T2 1 + 27274 T + p^{5} T^{2}
59 1−2490T+p5T2 1 - 2490 T + p^{5} T^{2}
61 1+11096T+p5T2 1 + 11096 T + p^{5} T^{2}
67 1−39756T+p5T2 1 - 39756 T + p^{5} T^{2}
71 1+69888T+p5T2 1 + 69888 T + p^{5} T^{2}
73 1−16450T+p5T2 1 - 16450 T + p^{5} T^{2}
79 1−78376T+p5T2 1 - 78376 T + p^{5} T^{2}
83 1−109818T+p5T2 1 - 109818 T + p^{5} T^{2}
89 1+56966T+p5T2 1 + 56966 T + p^{5} T^{2}
97 1+115946T+p5T2 1 + 115946 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

−14.13809402524531324416432590787, −13.52165000273312647350281224748, −12.37326325523858471208190566018, −10.19971052419984600160359067477, −9.563342109165115825828696484751, −8.116522669086459206795458627772, −7.44735158295488635814459990857, −5.05073151177499745827916939669, −3.15049277146873784004904647745, −1.98971742174572232888128510133, 1.98971742174572232888128510133, 3.15049277146873784004904647745, 5.05073151177499745827916939669, 7.44735158295488635814459990857, 8.116522669086459206795458627772, 9.563342109165115825828696484751, 10.19971052419984600160359067477, 12.37326325523858471208190566018, 13.52165000273312647350281224748, 14.13809402524531324416432590787

Graph of the ZZ-function along the critical line