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Properties

Label 2-52e2-1.1-c1-0-26
Degree 22
Conductor 27042704
Sign 11
Analytic cond. 21.591521.5915
Root an. cond. 4.646674.64667
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.76·3-s + 4.24·5-s + 2.37·7-s + 4.63·9-s + 0.178·11-s − 11.7·15-s + 1.03·17-s + 5.25·19-s − 6.56·21-s + 0.130·23-s + 13.0·25-s − 4.52·27-s + 8.33·29-s + 2.94·31-s − 0.494·33-s + 10.0·35-s − 6.03·37-s − 0.264·41-s − 0.564·43-s + 19.7·45-s − 10.9·47-s − 1.36·49-s − 2.87·51-s + 2.41·53-s + 0.760·55-s − 14.5·57-s + 7.35·59-s + ⋯
L(s)  = 1  − 1.59·3-s + 1.90·5-s + 0.897·7-s + 1.54·9-s + 0.0539·11-s − 3.03·15-s + 0.252·17-s + 1.20·19-s − 1.43·21-s + 0.0271·23-s + 2.61·25-s − 0.871·27-s + 1.54·29-s + 0.529·31-s − 0.0860·33-s + 1.70·35-s − 0.992·37-s − 0.0413·41-s − 0.0861·43-s + 2.93·45-s − 1.59·47-s − 0.194·49-s − 0.402·51-s + 0.332·53-s + 0.102·55-s − 1.92·57-s + 0.957·59-s + ⋯

Functional equation

Λ(s)=(2704s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(2704s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 27042704    =    24⋅1322^{4} \cdot 13^{2}
Sign: 11
Analytic conductor: 21.591521.5915
Root analytic conductor: 4.646674.64667
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 2704, ( :1/2), 1)(2,\ 2704,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.8986845841.898684584
L(12)L(\frac12) ≈\approx 1.8986845841.898684584
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
13 1 1
good3 1+2.76T+3T2 1 + 2.76T + 3T^{2}
5 1−4.24T+5T2 1 - 4.24T + 5T^{2}
7 1−2.37T+7T2 1 - 2.37T + 7T^{2}
11 1−0.178T+11T2 1 - 0.178T + 11T^{2}
17 1−1.03T+17T2 1 - 1.03T + 17T^{2}
19 1−5.25T+19T2 1 - 5.25T + 19T^{2}
23 1−0.130T+23T2 1 - 0.130T + 23T^{2}
29 1−8.33T+29T2 1 - 8.33T + 29T^{2}
31 1−2.94T+31T2 1 - 2.94T + 31T^{2}
37 1+6.03T+37T2 1 + 6.03T + 37T^{2}
41 1+0.264T+41T2 1 + 0.264T + 41T^{2}
43 1+0.564T+43T2 1 + 0.564T + 43T^{2}
47 1+10.9T+47T2 1 + 10.9T + 47T^{2}
53 1−2.41T+53T2 1 - 2.41T + 53T^{2}
59 1−7.35T+59T2 1 - 7.35T + 59T^{2}
61 1+5.06T+61T2 1 + 5.06T + 61T^{2}
67 1−1.58T+67T2 1 - 1.58T + 67T^{2}
71 1+10.3T+71T2 1 + 10.3T + 71T^{2}
73 1+14.3T+73T2 1 + 14.3T + 73T^{2}
79 1−7.26T+79T2 1 - 7.26T + 79T^{2}
83 1−12.9T+83T2 1 - 12.9T + 83T^{2}
89 1−6.92T+89T2 1 - 6.92T + 89T^{2}
97 1+3.63T+97T2 1 + 3.63T + 97T^{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

−9.023040954832512457164632680347, −8.053650654615968769488361210523, −6.90530367604280605867303404472, −6.41859426098273627616223350967, −5.63463201912198505609226676440, −5.15566355911287136077186686048, −4.63256367987969710671163544977, −2.96103997637910477344852033986, −1.71393808287302505058481695129, −1.04303182035722041540687006314, 1.04303182035722041540687006314, 1.71393808287302505058481695129, 2.96103997637910477344852033986, 4.63256367987969710671163544977, 5.15566355911287136077186686048, 5.63463201912198505609226676440, 6.41859426098273627616223350967, 6.90530367604280605867303404472, 8.053650654615968769488361210523, 9.023040954832512457164632680347

Graph of the ZZ-function along the critical line