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Properties

Label 2-2646-1.1-c1-0-23
Degree 22
Conductor 26462646
Sign 11
Analytic cond. 21.128421.1284
Root an. cond. 4.596564.59656
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-s + 4-s + 2·5-s + 8-s + 2·10-s − 2·11-s + 13-s + 16-s − 3·17-s + 4·19-s + 2·20-s − 2·22-s + 3·23-s − 25-s + 26-s + 7·29-s + 11·31-s + 32-s − 3·34-s − 6·37-s + 4·38-s + 2·40-s + 6·41-s + 43-s − 2·44-s + 3·46-s + 8·47-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.894·5-s + 0.353·8-s + 0.632·10-s − 0.603·11-s + 0.277·13-s + 1/4·16-s − 0.727·17-s + 0.917·19-s + 0.447·20-s − 0.426·22-s + 0.625·23-s − 1/5·25-s + 0.196·26-s + 1.29·29-s + 1.97·31-s + 0.176·32-s − 0.514·34-s − 0.986·37-s + 0.648·38-s + 0.316·40-s + 0.937·41-s + 0.152·43-s − 0.301·44-s + 0.442·46-s + 1.16·47-s + ⋯

Functional equation

Λ(s)=(2646s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(2646s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 26462646    =    2⋅33⋅722 \cdot 3^{3} \cdot 7^{2}
Sign: 11
Analytic conductor: 21.128421.1284
Root analytic conductor: 4.596564.59656
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 2646, ( :1/2), 1)(2,\ 2646,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 3.5093151523.509315152
L(12)L(\frac12) ≈\approx 3.5093151523.509315152
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−T 1 - T
3 1 1
7 1 1
good5 1−2T+pT2 1 - 2 T + p T^{2}
11 1+2T+pT2 1 + 2 T + p T^{2}
13 1−T+pT2 1 - T + p T^{2}
17 1+3T+pT2 1 + 3 T + p T^{2}
19 1−4T+pT2 1 - 4 T + p T^{2}
23 1−3T+pT2 1 - 3 T + p T^{2}
29 1−7T+pT2 1 - 7 T + p T^{2}
31 1−11T+pT2 1 - 11 T + p T^{2}
37 1+6T+pT2 1 + 6 T + p T^{2}
41 1−6T+pT2 1 - 6 T + p T^{2}
43 1−T+pT2 1 - T + p T^{2}
47 1−8T+pT2 1 - 8 T + p T^{2}
53 1−T+pT2 1 - T + p T^{2}
59 1+7T+pT2 1 + 7 T + p T^{2}
61 1−14T+pT2 1 - 14 T + p T^{2}
67 1+11T+pT2 1 + 11 T + p T^{2}
71 1+11T+pT2 1 + 11 T + p T^{2}
73 1−8T+pT2 1 - 8 T + p T^{2}
79 1−4T+pT2 1 - 4 T + p T^{2}
83 1+pT2 1 + p T^{2}
89 1−T+pT2 1 - T + p T^{2}
97 1+16T+pT2 1 + 16 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

−8.858491582708342556427559261956, −8.064645491147101931790045387521, −7.13790621913547938209696958174, −6.41371642316385572236262321411, −5.71712844391571482840343232501, −4.99614562130570219516186374288, −4.21966667354524075129226149758, −3.00004241953254677623600625591, −2.38283992470961307825516552466, −1.13028052123358941518074717677, 1.13028052123358941518074717677, 2.38283992470961307825516552466, 3.00004241953254677623600625591, 4.21966667354524075129226149758, 4.99614562130570219516186374288, 5.71712844391571482840343232501, 6.41371642316385572236262321411, 7.13790621913547938209696958174, 8.064645491147101931790045387521, 8.858491582708342556427559261956

Graph of the ZZ-function along the critical line