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Properties

Label 2-6336-1.1-c1-0-3
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  − 2.82·5-s − 0.828·7-s + 11-s − 4.82·13-s + 2·17-s − 2·19-s − 2.82·23-s + 3.00·25-s + 3.65·29-s + 9.65·31-s + 2.34·35-s − 7.65·37-s + 0.343·41-s − 0.343·43-s − 12.4·47-s − 6.31·49-s − 6.82·53-s − 2.82·55-s − 1.65·59-s − 3.17·61-s + 13.6·65-s − 11.3·67-s + 4.48·71-s − 13.3·73-s − 0.828·77-s + 4.82·79-s + 9.65·83-s + ⋯
L(s)  = 1  − 1.26·5-s − 0.313·7-s + 0.301·11-s − 1.33·13-s + 0.485·17-s − 0.458·19-s − 0.589·23-s + 0.600·25-s + 0.679·29-s + 1.73·31-s + 0.396·35-s − 1.25·37-s + 0.0535·41-s − 0.0523·43-s − 1.82·47-s − 0.901·49-s − 0.937·53-s − 0.381·55-s − 0.215·59-s − 0.406·61-s + 1.69·65-s − 1.38·67-s + 0.532·71-s − 1.55·73-s − 0.0944·77-s + 0.543·79-s + 1.05·83-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 0.80934404980.8093440498
L(12)L(\frac12) ≈\approx 0.80934404980.8093440498
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+2.82T+5T2 1 + 2.82T + 5T^{2}
7 1+0.828T+7T2 1 + 0.828T + 7T^{2}
13 1+4.82T+13T2 1 + 4.82T + 13T^{2}
17 1−2T+17T2 1 - 2T + 17T^{2}
19 1+2T+19T2 1 + 2T + 19T^{2}
23 1+2.82T+23T2 1 + 2.82T + 23T^{2}
29 1−3.65T+29T2 1 - 3.65T + 29T^{2}
31 1−9.65T+31T2 1 - 9.65T + 31T^{2}
37 1+7.65T+37T2 1 + 7.65T + 37T^{2}
41 1−0.343T+41T2 1 - 0.343T + 41T^{2}
43 1+0.343T+43T2 1 + 0.343T + 43T^{2}
47 1+12.4T+47T2 1 + 12.4T + 47T^{2}
53 1+6.82T+53T2 1 + 6.82T + 53T^{2}
59 1+1.65T+59T2 1 + 1.65T + 59T^{2}
61 1+3.17T+61T2 1 + 3.17T + 61T^{2}
67 1+11.3T+67T2 1 + 11.3T + 67T^{2}
71 1−4.48T+71T2 1 - 4.48T + 71T^{2}
73 1+13.3T+73T2 1 + 13.3T + 73T^{2}
79 1−4.82T+79T2 1 - 4.82T + 79T^{2}
83 1−9.65T+83T2 1 - 9.65T + 83T^{2}
89 1−16T+89T2 1 - 16T + 89T^{2}
97 1−5.31T+97T2 1 - 5.31T + 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

−7.972007944822836248574710012428, −7.45456544310963278037573254931, −6.66916883016553632340557607255, −6.08525708781090043354901226065, −4.77727849978647910191516636017, −4.62605260942949346626473062326, −3.51592438580234671701714218037, −2.99679383975078778404603549766, −1.84033092862646119575103451648, −0.45303433350119744858200838320, 0.45303433350119744858200838320, 1.84033092862646119575103451648, 2.99679383975078778404603549766, 3.51592438580234671701714218037, 4.62605260942949346626473062326, 4.77727849978647910191516636017, 6.08525708781090043354901226065, 6.66916883016553632340557607255, 7.45456544310963278037573254931, 7.972007944822836248574710012428

Graph of the ZZ-function along the critical line