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Properties

Label 2-81675-1.1-c1-0-17
Degree 22
Conductor 8167581675
Sign 11
Analytic cond. 652.178652.178
Root an. cond. 25.537725.5377
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  − 2·4-s − 7-s − 13-s + 4·16-s + 4·19-s + 2·28-s − 3·29-s − 31-s + 4·37-s − 6·41-s + 11·43-s − 9·47-s − 6·49-s + 2·52-s − 6·53-s + 6·59-s − 8·61-s − 8·64-s + 4·67-s + 12·71-s − 7·73-s − 8·76-s − 2·79-s + 12·83-s + 6·89-s + 91-s − 8·97-s + ⋯
L(s)  = 1  − 4-s − 0.377·7-s − 0.277·13-s + 16-s + 0.917·19-s + 0.377·28-s − 0.557·29-s − 0.179·31-s + 0.657·37-s − 0.937·41-s + 1.67·43-s − 1.31·47-s − 6/7·49-s + 0.277·52-s − 0.824·53-s + 0.781·59-s − 1.02·61-s − 64-s + 0.488·67-s + 1.42·71-s − 0.819·73-s − 0.917·76-s − 0.225·79-s + 1.31·83-s + 0.635·89-s + 0.104·91-s − 0.812·97-s + ⋯

Functional equation

Λ(s)=(81675s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 81675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(81675s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 81675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 8167581675    =    33⋅52⋅1123^{3} \cdot 5^{2} \cdot 11^{2}
Sign: 11
Analytic conductor: 652.178652.178
Root analytic conductor: 25.537725.5377
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 81675, ( :1/2), 1)(2,\ 81675,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.1679922741.167992274
L(12)L(\frac12) ≈\approx 1.1679922741.167992274
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)
bad3 1 1
5 1 1
11 1 1
good2 1+pT2 1 + p T^{2}
7 1+T+pT2 1 + T + p T^{2}
13 1+T+pT2 1 + T + p T^{2}
17 1+pT2 1 + p T^{2}
19 1−4T+pT2 1 - 4 T + p T^{2}
23 1+pT2 1 + p T^{2}
29 1+3T+pT2 1 + 3 T + p T^{2}
31 1+T+pT2 1 + T + p T^{2}
37 1−4T+pT2 1 - 4 T + p T^{2}
41 1+6T+pT2 1 + 6 T + p T^{2}
43 1−11T+pT2 1 - 11 T + p T^{2}
47 1+9T+pT2 1 + 9 T + p T^{2}
53 1+6T+pT2 1 + 6 T + p T^{2}
59 1−6T+pT2 1 - 6 T + p T^{2}
61 1+8T+pT2 1 + 8 T + p T^{2}
67 1−4T+pT2 1 - 4 T + p T^{2}
71 1−12T+pT2 1 - 12 T + p T^{2}
73 1+7T+pT2 1 + 7 T + p T^{2}
79 1+2T+pT2 1 + 2 T + p T^{2}
83 1−12T+pT2 1 - 12 T + p T^{2}
89 1−6T+pT2 1 - 6 T + p T^{2}
97 1+8T+pT2 1 + 8 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

−14.05446780125512, −13.39994656245470, −13.02719796527908, −12.61711408440890, −12.07636853002946, −11.53048029020602, −10.96596113525077, −10.40557002914571, −9.735247775989648, −9.520044818462051, −9.100813831632075, −8.375503920804214, −7.935734470083110, −7.448759416902168, −6.817278434224986, −6.126497368106461, −5.673182907678261, −4.967978312241981, −4.693258976998335, −3.859507980803551, −3.427130464487703, −2.843576248153046, −1.957384623225147, −1.154057297458743, −0.3946554975066199, 0.3946554975066199, 1.154057297458743, 1.957384623225147, 2.843576248153046, 3.427130464487703, 3.859507980803551, 4.693258976998335, 4.967978312241981, 5.673182907678261, 6.126497368106461, 6.817278434224986, 7.448759416902168, 7.935734470083110, 8.375503920804214, 9.100813831632075, 9.520044818462051, 9.735247775989648, 10.40557002914571, 10.96596113525077, 11.53048029020602, 12.07636853002946, 12.61711408440890, 13.02719796527908, 13.39994656245470, 14.05446780125512

Graph of the ZZ-function along the critical line