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Properties

Label 2-1638-1.1-c1-0-3
Degree 22
Conductor 16381638
Sign 11
Analytic cond. 13.079413.0794
Root an. cond. 3.616553.61655
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 − 4·5-s − 7-s + 8-s − 4·10-s + 11-s + 13-s − 14-s + 16-s − 4·17-s + 2·19-s − 4·20-s + 22-s + 7·23-s + 11·25-s + 26-s − 28-s + 8·29-s + 3·31-s + 32-s − 4·34-s + 4·35-s + 7·37-s + 2·38-s − 4·40-s + 7·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 1.78·5-s − 0.377·7-s + 0.353·8-s − 1.26·10-s + 0.301·11-s + 0.277·13-s − 0.267·14-s + 1/4·16-s − 0.970·17-s + 0.458·19-s − 0.894·20-s + 0.213·22-s + 1.45·23-s + 11/5·25-s + 0.196·26-s − 0.188·28-s + 1.48·29-s + 0.538·31-s + 0.176·32-s − 0.685·34-s + 0.676·35-s + 1.15·37-s + 0.324·38-s − 0.632·40-s + 1.09·41-s + ⋯

Functional equation

Λ(s)=(1638s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1638s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 16381638    =    2⋅32⋅7⋅132 \cdot 3^{2} \cdot 7 \cdot 13
Sign: 11
Analytic conductor: 13.079413.0794
Root analytic conductor: 3.616553.61655
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1638, ( :1/2), 1)(2,\ 1638,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.8397487371.839748737
L(12)L(\frac12) ≈\approx 1.8397487371.839748737
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+T 1 + T
13 1−T 1 - T
good5 1+4T+pT2 1 + 4 T + p T^{2}
11 1−T+pT2 1 - T + p T^{2}
17 1+4T+pT2 1 + 4 T + p T^{2}
19 1−2T+pT2 1 - 2 T + p T^{2}
23 1−7T+pT2 1 - 7 T + p T^{2}
29 1−8T+pT2 1 - 8 T + p T^{2}
31 1−3T+pT2 1 - 3 T + p T^{2}
37 1−7T+pT2 1 - 7 T + p T^{2}
41 1−7T+pT2 1 - 7 T + p T^{2}
43 1+8T+pT2 1 + 8 T + p T^{2}
47 1+3T+pT2 1 + 3 T + p T^{2}
53 1+pT2 1 + p T^{2}
59 1−6T+pT2 1 - 6 T + p T^{2}
61 1+13T+pT2 1 + 13 T + p T^{2}
67 1−7T+pT2 1 - 7 T + p T^{2}
71 1+4T+pT2 1 + 4 T + p T^{2}
73 1−9T+pT2 1 - 9 T + p T^{2}
79 1+13T+pT2 1 + 13 T + p T^{2}
83 1−16T+pT2 1 - 16 T + p T^{2}
89 1−6T+pT2 1 - 6 T + p T^{2}
97 1−11T+pT2 1 - 11 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

−9.229708445181394321480923526929, −8.446510046834741944724142023215, −7.70223365400781146351389504873, −6.89743886981333030203070546273, −6.32586188918121275548015521553, −4.91521481773069545833368645820, −4.38408648015283815906866115610, −3.48622278921391586115492321920, −2.77045286705004577101314310504, −0.859680047349367001421384183810, 0.859680047349367001421384183810, 2.77045286705004577101314310504, 3.48622278921391586115492321920, 4.38408648015283815906866115610, 4.91521481773069545833368645820, 6.32586188918121275548015521553, 6.89743886981333030203070546273, 7.70223365400781146351389504873, 8.446510046834741944724142023215, 9.229708445181394321480923526929

Graph of the ZZ-function along the critical line