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Properties

Label 2-87-87.86-c0-0-1
Degree 22
Conductor 8787
Sign 11
Analytic cond. 0.04341860.0434186
Root an. cond. 0.2083710.208371
Motivic weight 00
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 − 3-s − 6-s − 7-s − 8-s + 9-s + 11-s − 13-s − 14-s − 16-s + 17-s + 18-s + 21-s + 22-s + 24-s + 25-s − 26-s − 27-s − 29-s − 33-s + 34-s + 39-s − 2·41-s + 42-s + 47-s + 48-s + 50-s + ⋯
L(s)  = 1  + 2-s − 3-s − 6-s − 7-s − 8-s + 9-s + 11-s − 13-s − 14-s − 16-s + 17-s + 18-s + 21-s + 22-s + 24-s + 25-s − 26-s − 27-s − 29-s − 33-s + 34-s + 39-s − 2·41-s + 42-s + 47-s + 48-s + 50-s + ⋯

Functional equation

Λ(s)=(87s/2 ΓC(s) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
Λ(s)=(87s/2 ΓC(s) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 8787    =    3⋅293 \cdot 29
Sign: 11
Analytic conductor: 0.04341860.0434186
Root analytic conductor: 0.2083710.208371
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: χ87(86,⋅)\chi_{87} (86, \cdot )
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 87, ( :0), 1)(2,\ 87,\ (\ :0),\ 1)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.57705863700.5770586370
L(12)L(\frac12) ≈\approx 0.57705863700.5770586370
L(1)L(1) 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)
bad3 1+T 1 + T
29 1+T 1 + T
good2 1−T+T2 1 - T + T^{2}
5 (1−T)(1+T) ( 1 - T )( 1 + T )
7 1+T+T2 1 + T + T^{2}
11 1−T+T2 1 - T + T^{2}
13 1+T+T2 1 + T + T^{2}
17 1−T+T2 1 - T + T^{2}
19 (1−T)(1+T) ( 1 - T )( 1 + T )
23 (1−T)(1+T) ( 1 - T )( 1 + T )
31 (1−T)(1+T) ( 1 - T )( 1 + T )
37 (1−T)(1+T) ( 1 - T )( 1 + T )
41 (1+T)2 ( 1 + T )^{2}
43 (1−T)(1+T) ( 1 - T )( 1 + T )
47 1−T+T2 1 - T + T^{2}
53 (1−T)(1+T) ( 1 - T )( 1 + T )
59 (1−T)(1+T) ( 1 - T )( 1 + T )
61 (1−T)(1+T) ( 1 - T )( 1 + T )
67 1+T+T2 1 + T + T^{2}
71 (1−T)(1+T) ( 1 - T )( 1 + T )
73 (1−T)(1+T) ( 1 - T )( 1 + T )
79 (1−T)(1+T) ( 1 - T )( 1 + T )
83 (1−T)(1+T) ( 1 - T )( 1 + T )
89 1−T+T2 1 - T + T^{2}
97 (1−T)(1+T) ( 1 - T )( 1 + T )
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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

−14.39892195706409998885986607529, −13.18133367873374302632506132995, −12.37546425848442860075315832869, −11.74152542706301551240275566622, −10.16119846875737190734833337722, −9.215490822664834597853334075143, −7.06152498257571751488998986929, −6.04035672910010770144714336001, −4.91310283918857793479423733862, −3.53215773738360323132247593606, 3.53215773738360323132247593606, 4.91310283918857793479423733862, 6.04035672910010770144714336001, 7.06152498257571751488998986929, 9.215490822664834597853334075143, 10.16119846875737190734833337722, 11.74152542706301551240275566622, 12.37546425848442860075315832869, 13.18133367873374302632506132995, 14.39892195706409998885986607529

Graph of the ZZ-function along the critical line