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Properties

Label 2-6336-1.1-c1-0-10
Degree 22
Conductor 63366336
Sign 11
Analytic cond. 50.593250.5932
Root an. cond. 7.112897.11289
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  − 0.561·5-s − 11-s − 2·13-s − 7.12·17-s − 1.12·19-s − 7.68·23-s − 4.68·25-s + 7.12·29-s + 5.43·31-s + 5.68·37-s + 8.24·41-s − 1.12·43-s − 4·47-s − 7·49-s + 8.24·53-s + 0.561·55-s + 0.315·59-s − 9.36·61-s + 1.12·65-s + 7.68·67-s + 15.6·71-s − 6·73-s + 13.1·79-s + 11.3·83-s + 4·85-s − 0.561·89-s + 0.630·95-s + ⋯
L(s)  = 1  − 0.251·5-s − 0.301·11-s − 0.554·13-s − 1.72·17-s − 0.257·19-s − 1.60·23-s − 0.936·25-s + 1.32·29-s + 0.976·31-s + 0.934·37-s + 1.28·41-s − 0.171·43-s − 0.583·47-s − 49-s + 1.13·53-s + 0.0757·55-s + 0.0410·59-s − 1.19·61-s + 0.139·65-s + 0.938·67-s + 1.86·71-s − 0.702·73-s + 1.47·79-s + 1.24·83-s + 0.433·85-s − 0.0595·89-s + 0.0647·95-s + ⋯

Functional equation

Λ(s)=(6336s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(6336s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 63366336    =    26⋅32⋅112^{6} \cdot 3^{2} \cdot 11
Sign: 11
Analytic conductor: 50.593250.5932
Root analytic conductor: 7.112897.11289
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 6336, ( :1/2), 1)(2,\ 6336,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.2111336111.211133611
L(12)L(\frac12) ≈\approx 1.2111336111.211133611
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
3 1 1
11 1+T 1 + T
good5 1+0.561T+5T2 1 + 0.561T + 5T^{2}
7 1+7T2 1 + 7T^{2}
13 1+2T+13T2 1 + 2T + 13T^{2}
17 1+7.12T+17T2 1 + 7.12T + 17T^{2}
19 1+1.12T+19T2 1 + 1.12T + 19T^{2}
23 1+7.68T+23T2 1 + 7.68T + 23T^{2}
29 1−7.12T+29T2 1 - 7.12T + 29T^{2}
31 1−5.43T+31T2 1 - 5.43T + 31T^{2}
37 1−5.68T+37T2 1 - 5.68T + 37T^{2}
41 1−8.24T+41T2 1 - 8.24T + 41T^{2}
43 1+1.12T+43T2 1 + 1.12T + 43T^{2}
47 1+4T+47T2 1 + 4T + 47T^{2}
53 1−8.24T+53T2 1 - 8.24T + 53T^{2}
59 1−0.315T+59T2 1 - 0.315T + 59T^{2}
61 1+9.36T+61T2 1 + 9.36T + 61T^{2}
67 1−7.68T+67T2 1 - 7.68T + 67T^{2}
71 1−15.6T+71T2 1 - 15.6T + 71T^{2}
73 1+6T+73T2 1 + 6T + 73T^{2}
79 1−13.1T+79T2 1 - 13.1T + 79T^{2}
83 1−11.3T+83T2 1 - 11.3T + 83T^{2}
89 1+0.561T+89T2 1 + 0.561T + 89T^{2}
97 1−5.68T+97T2 1 - 5.68T + 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

−8.019430595467747593397337984093, −7.46803125053773855341117261493, −6.33428244849035194162063066223, −6.27417661180688312398336058062, −4.98215324249077053170380155304, −4.45555337512782494579671955400, −3.74790363952911656594424461630, −2.56693275154179492329905877024, −2.06308837701462828438497134662, −0.54421452148532331857859714146, 0.54421452148532331857859714146, 2.06308837701462828438497134662, 2.56693275154179492329905877024, 3.74790363952911656594424461630, 4.45555337512782494579671955400, 4.98215324249077053170380155304, 6.27417661180688312398336058062, 6.33428244849035194162063066223, 7.46803125053773855341117261493, 8.019430595467747593397337984093

Graph of the ZZ-function along the critical line