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Properties

Label 2-87856-1.1-c1-0-0
Degree 22
Conductor 8785687856
Sign 11
Analytic cond. 701.533701.533
Root an. cond. 26.486426.4864
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  + 3-s − 7-s − 2·9-s − 6·11-s + 5·13-s − 19-s − 21-s + 3·23-s − 5·25-s − 5·27-s − 9·29-s − 4·31-s − 6·33-s − 2·37-s + 5·39-s − 8·43-s − 6·49-s − 3·53-s − 57-s − 9·59-s + 10·61-s + 2·63-s − 5·67-s + 3·69-s − 6·71-s + 7·73-s − 5·75-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.377·7-s − 2/3·9-s − 1.80·11-s + 1.38·13-s − 0.229·19-s − 0.218·21-s + 0.625·23-s − 25-s − 0.962·27-s − 1.67·29-s − 0.718·31-s − 1.04·33-s − 0.328·37-s + 0.800·39-s − 1.21·43-s − 6/7·49-s − 0.412·53-s − 0.132·57-s − 1.17·59-s + 1.28·61-s + 0.251·63-s − 0.610·67-s + 0.361·69-s − 0.712·71-s + 0.819·73-s − 0.577·75-s + ⋯

Functional equation

Λ(s)=(87856s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 87856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(87856s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 87856 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 8785687856    =    24⋅172⋅192^{4} \cdot 17^{2} \cdot 19
Sign: 11
Analytic conductor: 701.533701.533
Root analytic conductor: 26.486426.4864
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 87856, ( :1/2), 1)(2,\ 87856,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.29168007490.2916800749
L(12)L(\frac12) ≈\approx 0.29168007490.2916800749
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
17 1 1
19 1+T 1 + T
good3 1−T+pT2 1 - T + p T^{2}
5 1+pT2 1 + p T^{2}
7 1+T+pT2 1 + T + p T^{2}
11 1+6T+pT2 1 + 6 T + p T^{2}
13 1−5T+pT2 1 - 5 T + p T^{2}
23 1−3T+pT2 1 - 3 T + p T^{2}
29 1+9T+pT2 1 + 9 T + p T^{2}
31 1+4T+pT2 1 + 4 T + 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+pT2 1 + p T^{2}
53 1+3T+pT2 1 + 3 T + p T^{2}
59 1+9T+pT2 1 + 9 T + p T^{2}
61 1−10T+pT2 1 - 10 T + p T^{2}
67 1+5T+pT2 1 + 5 T + p T^{2}
71 1+6T+pT2 1 + 6 T + p T^{2}
73 1−7T+pT2 1 - 7 T + p T^{2}
79 1+10T+pT2 1 + 10 T + p T^{2}
83 1−6T+pT2 1 - 6 T + p T^{2}
89 1+12T+pT2 1 + 12 T + p T^{2}
97 1−10T+pT2 1 - 10 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

−13.69689906603957, −13.41405686662163, −12.99909316590978, −12.73793980744086, −11.81220130213798, −11.31533831000028, −10.94538926117473, −10.48317618935484, −9.867894765091065, −9.319502179078990, −8.857311781669834, −8.309626178765610, −7.908262215570247, −7.495661450785631, −6.765745365478681, −6.114148312313132, −5.559390212451067, −5.311830533242366, −4.458789815333955, −3.581554877743056, −3.405007963317238, −2.717076622959414, −2.074919925400632, −1.457841846141822, −0.1561256662306693, 0.1561256662306693, 1.457841846141822, 2.074919925400632, 2.717076622959414, 3.405007963317238, 3.581554877743056, 4.458789815333955, 5.311830533242366, 5.559390212451067, 6.114148312313132, 6.765745365478681, 7.495661450785631, 7.908262215570247, 8.309626178765610, 8.857311781669834, 9.319502179078990, 9.867894765091065, 10.48317618935484, 10.94538926117473, 11.31533831000028, 11.81220130213798, 12.73793980744086, 12.99909316590978, 13.41405686662163, 13.69689906603957

Graph of the ZZ-function along the critical line