↑

Properties

Label 2-2646-1.1-c1-0-25
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 3·5-s − 8-s − 3·10-s + 3·11-s + 4·13-s + 16-s + 6·17-s + 7·19-s + 3·20-s − 3·22-s − 3·23-s + 4·25-s − 4·26-s − 5·31-s − 32-s − 6·34-s − 7·37-s − 7·38-s − 3·40-s + 9·41-s − 10·43-s + 3·44-s + 3·46-s − 6·47-s − 4·50-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 1.34·5-s − 0.353·8-s − 0.948·10-s + 0.904·11-s + 1.10·13-s + 1/4·16-s + 1.45·17-s + 1.60·19-s + 0.670·20-s − 0.639·22-s − 0.625·23-s + 4/5·25-s − 0.784·26-s − 0.898·31-s − 0.176·32-s − 1.02·34-s − 1.15·37-s − 1.13·38-s − 0.474·40-s + 1.40·41-s − 1.52·43-s + 0.452·44-s + 0.442·46-s − 0.875·47-s − 0.565·50-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 2.0939263422.093926342
L(12)L(\frac12) ≈\approx 2.0939263422.093926342
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 1−3T+pT2 1 - 3 T + p T^{2}
11 1−3T+pT2 1 - 3 T + p T^{2}
13 1−4T+pT2 1 - 4 T + p T^{2}
17 1−6T+pT2 1 - 6 T + p T^{2}
19 1−7T+pT2 1 - 7 T + p T^{2}
23 1+3T+pT2 1 + 3 T + p T^{2}
29 1+pT2 1 + p T^{2}
31 1+5T+pT2 1 + 5 T + p T^{2}
37 1+7T+pT2 1 + 7 T + p T^{2}
41 1−9T+pT2 1 - 9 T + p T^{2}
43 1+10T+pT2 1 + 10 T + p T^{2}
47 1+6T+pT2 1 + 6 T + p T^{2}
53 1−12T+pT2 1 - 12 T + p T^{2}
59 1−6T+pT2 1 - 6 T + p T^{2}
61 1+8T+pT2 1 + 8 T + p T^{2}
67 1+4T+pT2 1 + 4 T + p T^{2}
71 1−9T+pT2 1 - 9 T + p T^{2}
73 1+2T+pT2 1 + 2 T + p T^{2}
79 1+10T+pT2 1 + 10 T + p T^{2}
83 1+pT2 1 + p T^{2}
89 1+15T+pT2 1 + 15 T + p T^{2}
97 1+8T+pT2 1 + 8 T + p T^{2}
show more
show less
   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

−8.992251307828531296445343338023, −8.250819661564719776627888855362, −7.34475786918817202650312765182, −6.59247589213525769518809192749, −5.73304100710876992030851819199, −5.41376865336615198247524157398, −3.83254249011259881446516091298, −3.04507790741923758224766618955, −1.72414285314126353169175309195, −1.16041265383789495288315554440, 1.16041265383789495288315554440, 1.72414285314126353169175309195, 3.04507790741923758224766618955, 3.83254249011259881446516091298, 5.41376865336615198247524157398, 5.73304100710876992030851819199, 6.59247589213525769518809192749, 7.34475786918817202650312765182, 8.250819661564719776627888855362, 8.992251307828531296445343338023

Graph of the ZZ-function along the critical line