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Properties

Label 2-2646-1.1-c1-0-15
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 + 4·5-s − 8-s − 4·10-s + 4·11-s − 3·13-s + 16-s − 7·17-s − 2·19-s + 4·20-s − 4·22-s + 23-s + 11·25-s + 3·26-s − 29-s + 9·31-s − 32-s + 7·34-s + 2·37-s + 2·38-s − 4·40-s + 6·41-s + 11·43-s + 4·44-s − 46-s − 6·47-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 1.78·5-s − 0.353·8-s − 1.26·10-s + 1.20·11-s − 0.832·13-s + 1/4·16-s − 1.69·17-s − 0.458·19-s + 0.894·20-s − 0.852·22-s + 0.208·23-s + 11/5·25-s + 0.588·26-s − 0.185·29-s + 1.61·31-s − 0.176·32-s + 1.20·34-s + 0.328·37-s + 0.324·38-s − 0.632·40-s + 0.937·41-s + 1.67·43-s + 0.603·44-s − 0.147·46-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 1.9093444151.909344415
L(12)L(\frac12) ≈\approx 1.9093444151.909344415
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 1−4T+pT2 1 - 4 T + p T^{2}
11 1−4T+pT2 1 - 4 T + p T^{2}
13 1+3T+pT2 1 + 3 T + p T^{2}
17 1+7T+pT2 1 + 7 T + p T^{2}
19 1+2T+pT2 1 + 2 T + p T^{2}
23 1−T+pT2 1 - T + p T^{2}
29 1+T+pT2 1 + T + p T^{2}
31 1−9T+pT2 1 - 9 T + p T^{2}
37 1−2T+pT2 1 - 2 T + p T^{2}
41 1−6T+pT2 1 - 6 T + p T^{2}
43 1−11T+pT2 1 - 11 T + p T^{2}
47 1+6T+pT2 1 + 6 T + p T^{2}
53 1−9T+pT2 1 - 9 T + p T^{2}
59 1+5T+pT2 1 + 5 T + p T^{2}
61 1−6T+pT2 1 - 6 T + p T^{2}
67 1−7T+pT2 1 - 7 T + p T^{2}
71 1−7T+pT2 1 - 7 T + p T^{2}
73 1−14T+pT2 1 - 14 T + p T^{2}
79 1+6T+pT2 1 + 6 T + p T^{2}
83 1+4T+pT2 1 + 4 T + p T^{2}
89 1+3T+pT2 1 + 3 T + p T^{2}
97 1−8T+pT2 1 - 8 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

−9.076386420504345035537011996118, −8.386834803015030063906818806717, −7.17230892063836621876590463825, −6.49244247951320148958731343293, −6.12230164783542580596662547870, −5.05352817881679124240654755585, −4.16226096911516495331985392522, −2.57898691887524846597709635063, −2.13967823750617304276182350687, −0.998858625839110032845229095752, 0.998858625839110032845229095752, 2.13967823750617304276182350687, 2.57898691887524846597709635063, 4.16226096911516495331985392522, 5.05352817881679124240654755585, 6.12230164783542580596662547870, 6.49244247951320148958731343293, 7.17230892063836621876590463825, 8.386834803015030063906818806717, 9.076386420504345035537011996118

Graph of the ZZ-function along the critical line