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Properties

Label 2-3276-1.1-c1-0-21
Degree 22
Conductor 32763276
Sign −1-1
Analytic cond. 26.158926.1589
Root an. cond. 5.114585.11458
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.73·5-s + 7-s − 4.73·11-s + 13-s + 2.19·17-s + 7.19·19-s − 3·23-s − 2.00·25-s − 0.464·29-s + 1.19·31-s − 1.73·35-s + 4.19·37-s − 3.46·41-s − 7·43-s − 7.73·47-s + 49-s + 9.92·53-s + 8.19·55-s + 10.3·59-s − 10·61-s − 1.73·65-s − 14.3·67-s + 1.26·71-s − 9.19·73-s − 4.73·77-s − 11.3·79-s + 0.803·83-s + ⋯
L(s)  = 1  − 0.774·5-s + 0.377·7-s − 1.42·11-s + 0.277·13-s + 0.532·17-s + 1.65·19-s − 0.625·23-s − 0.400·25-s − 0.0861·29-s + 0.214·31-s − 0.292·35-s + 0.689·37-s − 0.541·41-s − 1.06·43-s − 1.12·47-s + 0.142·49-s + 1.36·53-s + 1.10·55-s + 1.35·59-s − 1.28·61-s − 0.214·65-s − 1.75·67-s + 0.150·71-s − 1.07·73-s − 0.539·77-s − 1.28·79-s + 0.0882·83-s + ⋯

Functional equation

Λ(s)=(3276s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(3276s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 32763276    =    22⋅32⋅7⋅132^{2} \cdot 3^{2} \cdot 7 \cdot 13
Sign: −1-1
Analytic conductor: 26.158926.1589
Root analytic conductor: 5.114585.11458
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 3276, ( :1/2), −1)(2,\ 3276,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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
7 1−T 1 - T
13 1−T 1 - T
good5 1+1.73T+5T2 1 + 1.73T + 5T^{2}
11 1+4.73T+11T2 1 + 4.73T + 11T^{2}
17 1−2.19T+17T2 1 - 2.19T + 17T^{2}
19 1−7.19T+19T2 1 - 7.19T + 19T^{2}
23 1+3T+23T2 1 + 3T + 23T^{2}
29 1+0.464T+29T2 1 + 0.464T + 29T^{2}
31 1−1.19T+31T2 1 - 1.19T + 31T^{2}
37 1−4.19T+37T2 1 - 4.19T + 37T^{2}
41 1+3.46T+41T2 1 + 3.46T + 41T^{2}
43 1+7T+43T2 1 + 7T + 43T^{2}
47 1+7.73T+47T2 1 + 7.73T + 47T^{2}
53 1−9.92T+53T2 1 - 9.92T + 53T^{2}
59 1−10.3T+59T2 1 - 10.3T + 59T^{2}
61 1+10T+61T2 1 + 10T + 61T^{2}
67 1+14.3T+67T2 1 + 14.3T + 67T^{2}
71 1−1.26T+71T2 1 - 1.26T + 71T^{2}
73 1+9.19T+73T2 1 + 9.19T + 73T^{2}
79 1+11.3T+79T2 1 + 11.3T + 79T^{2}
83 1−0.803T+83T2 1 - 0.803T + 83T^{2}
89 1+11.1T+89T2 1 + 11.1T + 89T^{2}
97 1−2.80T+97T2 1 - 2.80T + 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

−7.999707871184336575170437022828, −7.76255328296783370355831283802, −6.99638639380194799722392052601, −5.79626303552497990683709708201, −5.26665880898826083530694287613, −4.39919426586222085341056702165, −3.45827785826426306683713689530, −2.71043422278831614548573942804, −1.39839987249752754708912214909, 0, 1.39839987249752754708912214909, 2.71043422278831614548573942804, 3.45827785826426306683713689530, 4.39919426586222085341056702165, 5.26665880898826083530694287613, 5.79626303552497990683709708201, 6.99638639380194799722392052601, 7.76255328296783370355831283802, 7.999707871184336575170437022828

Graph of the ZZ-function along the critical line