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Properties

Label 2-1083-3.2-c0-0-1
Degree 22
Conductor 10831083
Sign 11
Analytic cond. 0.5404870.540487
Root an. cond. 0.7351780.735178
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  + 3-s + 4-s − 7-s + 9-s + 12-s − 13-s + 16-s − 21-s + 25-s + 27-s − 28-s − 31-s + 36-s − 37-s − 39-s − 43-s + 48-s − 52-s − 61-s − 63-s + 64-s − 67-s − 73-s + 75-s − 79-s + 81-s − 84-s + ⋯
L(s)  = 1  + 3-s + 4-s − 7-s + 9-s + 12-s − 13-s + 16-s − 21-s + 25-s + 27-s − 28-s − 31-s + 36-s − 37-s − 39-s − 43-s + 48-s − 52-s − 61-s − 63-s + 64-s − 67-s − 73-s + 75-s − 79-s + 81-s − 84-s + ⋯

Functional equation

Λ(s)=(1083s/2 ΓC(s) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1083 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
Λ(s)=(1083s/2 ΓC(s) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1083 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 10831083    =    3⋅1923 \cdot 19^{2}
Sign: 11
Analytic conductor: 0.5404870.540487
Root analytic conductor: 0.7351780.735178
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: χ1083(362,⋅)\chi_{1083} (362, \cdot )
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1083, ( :0), 1)(2,\ 1083,\ (\ :0),\ 1)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.5932318531.593231853
L(12)L(\frac12) ≈\approx 1.5932318531.593231853
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
19 1 1
good2 (1−T)(1+T) ( 1 - T )( 1 + T )
5 (1−T)(1+T) ( 1 - T )( 1 + T )
7 1+T+T2 1 + T + T^{2}
11 (1−T)(1+T) ( 1 - T )( 1 + T )
13 1+T+T2 1 + T + T^{2}
17 (1−T)(1+T) ( 1 - T )( 1 + T )
23 (1−T)(1+T) ( 1 - T )( 1 + T )
29 (1−T)(1+T) ( 1 - T )( 1 + T )
31 1+T+T2 1 + T + T^{2}
37 1+T+T2 1 + T + T^{2}
41 (1−T)(1+T) ( 1 - T )( 1 + T )
43 1+T+T2 1 + T + T^{2}
47 (1−T)(1+T) ( 1 - T )( 1 + T )
53 (1−T)(1+T) ( 1 - T )( 1 + T )
59 (1−T)(1+T) ( 1 - T )( 1 + T )
61 1+T+T2 1 + T + T^{2}
67 1+T+T2 1 + T + T^{2}
71 (1−T)(1+T) ( 1 - T )( 1 + T )
73 1+T+T2 1 + T + T^{2}
79 1+T+T2 1 + T + T^{2}
83 (1−T)(1+T) ( 1 - T )( 1 + T )
89 (1−T)(1+T) ( 1 - T )( 1 + T )
97 (1−T)2 ( 1 - 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

−10.06290955317929830767358687855, −9.288797674257555469372289811718, −8.452967929121358153835405476936, −7.35157812485478151889927165723, −7.02532427698434232535483449218, −6.05528727277577329109096119249, −4.80679393526692214256010234380, −3.45799166917935983993729646004, −2.86640946537598527360617691881, −1.79957236321509578091517513370, 1.79957236321509578091517513370, 2.86640946537598527360617691881, 3.45799166917935983993729646004, 4.80679393526692214256010234380, 6.05528727277577329109096119249, 7.02532427698434232535483449218, 7.35157812485478151889927165723, 8.452967929121358153835405476936, 9.288797674257555469372289811718, 10.06290955317929830767358687855

Graph of the ZZ-function along the critical line