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Properties

Label 2-6336-1.1-c1-0-17
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 − 3.46·7-s + 11-s − 6.89·13-s − 6.29·17-s + 6.29·19-s − 4.89·23-s + 3.00·25-s − 0.635·29-s + 5.65·31-s − 9.79·35-s + 7.79·37-s + 0.635·41-s − 0.635·43-s + 8.89·47-s + 4.99·49-s − 9.75·53-s + 2.82·55-s + 13.7·59-s − 1.10·61-s − 19.5·65-s − 0.898·71-s + 6·73-s − 3.46·77-s + 16.0·79-s + 13.7·83-s − 17.7·85-s + ⋯
L(s)  = 1  + 1.26·5-s − 1.30·7-s + 0.301·11-s − 1.91·13-s − 1.52·17-s + 1.44·19-s − 1.02·23-s + 0.600·25-s − 0.118·29-s + 1.01·31-s − 1.65·35-s + 1.28·37-s + 0.0992·41-s − 0.0969·43-s + 1.29·47-s + 0.714·49-s − 1.34·53-s + 0.381·55-s + 1.79·59-s − 0.140·61-s − 2.42·65-s − 0.106·71-s + 0.702·73-s − 0.394·77-s + 1.80·79-s + 1.51·83-s − 1.93·85-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 1.6819505551.681950555
L(12)L(\frac12) ≈\approx 1.6819505551.681950555
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+3.46T+7T2 1 + 3.46T + 7T^{2}
13 1+6.89T+13T2 1 + 6.89T + 13T^{2}
17 1+6.29T+17T2 1 + 6.29T + 17T^{2}
19 1−6.29T+19T2 1 - 6.29T + 19T^{2}
23 1+4.89T+23T2 1 + 4.89T + 23T^{2}
29 1+0.635T+29T2 1 + 0.635T + 29T^{2}
31 1−5.65T+31T2 1 - 5.65T + 31T^{2}
37 1−7.79T+37T2 1 - 7.79T + 37T^{2}
41 1−0.635T+41T2 1 - 0.635T + 41T^{2}
43 1+0.635T+43T2 1 + 0.635T + 43T^{2}
47 1−8.89T+47T2 1 - 8.89T + 47T^{2}
53 1+9.75T+53T2 1 + 9.75T + 53T^{2}
59 1−13.7T+59T2 1 - 13.7T + 59T^{2}
61 1+1.10T+61T2 1 + 1.10T + 61T^{2}
67 1+67T2 1 + 67T^{2}
71 1+0.898T+71T2 1 + 0.898T + 71T^{2}
73 1−6T+73T2 1 - 6T + 73T^{2}
79 1−16.0T+79T2 1 - 16.0T + 79T^{2}
83 1−13.7T+83T2 1 - 13.7T + 83T^{2}
89 1−11.3T+89T2 1 - 11.3T + 89T^{2}
97 1−10T+97T2 1 - 10T + 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

−7.905223123986246862873064002210, −7.21439262108034744135066405181, −6.45875311241772379344711071998, −6.11157619207806244802977224557, −5.20916201261149202982237403020, −4.54955986767053330863450320982, −3.50014821287720639728779437694, −2.50445373908225859169491463917, −2.17559730464917952259053966265, −0.63956868108358842487663114790, 0.63956868108358842487663114790, 2.17559730464917952259053966265, 2.50445373908225859169491463917, 3.50014821287720639728779437694, 4.54955986767053330863450320982, 5.20916201261149202982237403020, 6.11157619207806244802977224557, 6.45875311241772379344711071998, 7.21439262108034744135066405181, 7.905223123986246862873064002210

Graph of the ZZ-function along the critical line