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Properties

Label 2-2646-1.1-c1-0-0
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 − 5·11-s − 6·13-s + 16-s − 4·17-s + 4·19-s − 2·20-s + 5·22-s + 4·23-s − 25-s + 6·26-s − 7·29-s − 3·31-s − 32-s + 4·34-s + 8·37-s − 4·38-s + 2·40-s − 6·41-s + 8·43-s − 5·44-s − 4·46-s + 6·47-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.894·5-s − 0.353·8-s + 0.632·10-s − 1.50·11-s − 1.66·13-s + 1/4·16-s − 0.970·17-s + 0.917·19-s − 0.447·20-s + 1.06·22-s + 0.834·23-s − 1/5·25-s + 1.17·26-s − 1.29·29-s − 0.538·31-s − 0.176·32-s + 0.685·34-s + 1.31·37-s − 0.648·38-s + 0.316·40-s − 0.937·41-s + 1.21·43-s − 0.753·44-s − 0.589·46-s + 0.875·47-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 0.49494340290.4949434029
L(12)L(\frac12) ≈\approx 0.49494340290.4949434029
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+5T+pT2 1 + 5 T + p T^{2}
13 1+6T+pT2 1 + 6 T + p T^{2}
17 1+4T+pT2 1 + 4 T + p T^{2}
19 1−4T+pT2 1 - 4 T + p T^{2}
23 1−4T+pT2 1 - 4 T + p T^{2}
29 1+7T+pT2 1 + 7 T + p T^{2}
31 1+3T+pT2 1 + 3 T + p T^{2}
37 1−8T+pT2 1 - 8 T + p T^{2}
41 1+6T+pT2 1 + 6 T + p T^{2}
43 1−8T+pT2 1 - 8 T + p T^{2}
47 1−6T+pT2 1 - 6 T + p T^{2}
53 1−6T+pT2 1 - 6 T + p T^{2}
59 1−7T+pT2 1 - 7 T + p T^{2}
61 1+pT2 1 + p T^{2}
67 1−10T+pT2 1 - 10 T + p T^{2}
71 1−4T+pT2 1 - 4 T + p T^{2}
73 1+13T+pT2 1 + 13 T + p T^{2}
79 1+3T+pT2 1 + 3 T + p T^{2}
83 1+7T+pT2 1 + 7 T + p T^{2}
89 1−6T+pT2 1 - 6 T + p T^{2}
97 1−5T+pT2 1 - 5 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

−8.852664685192944845022374486582, −7.948678631823869577486989243928, −7.41391959595930556776941898851, −7.08016768845276887296355816735, −5.67437679291396188991469247065, −5.02954632779517689480749219066, −4.05691084797417069928036854936, −2.87793724984651026489098410664, −2.21200891620018685371903520748, −0.45812635244474710229041310149, 0.45812635244474710229041310149, 2.21200891620018685371903520748, 2.87793724984651026489098410664, 4.05691084797417069928036854936, 5.02954632779517689480749219066, 5.67437679291396188991469247065, 7.08016768845276887296355816735, 7.41391959595930556776941898851, 7.948678631823869577486989243928, 8.852664685192944845022374486582

Graph of the ZZ-function along the critical line