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Properties

Label 2-3192-3192.797-c0-0-23
Degree 22
Conductor 31923192
Sign 11
Analytic cond. 1.593011.59301
Root an. cond. 1.262141.26214
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 + 4-s + 6-s − 7-s + 8-s + 9-s + 12-s − 14-s + 16-s + 18-s − 19-s − 21-s + 24-s + 25-s + 27-s − 28-s + 32-s + 36-s − 38-s − 42-s + 48-s + 49-s + 50-s + 54-s − 56-s − 57-s + ⋯
L(s)  = 1  + 2-s + 3-s + 4-s + 6-s − 7-s + 8-s + 9-s + 12-s − 14-s + 16-s + 18-s − 19-s − 21-s + 24-s + 25-s + 27-s − 28-s + 32-s + 36-s − 38-s − 42-s + 48-s + 49-s + 50-s + 54-s − 56-s − 57-s + ⋯

Functional equation

Λ(s)=(3192s/2 ΓC(s) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
Λ(s)=(3192s/2 ΓC(s) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 31923192    =    23⋅3⋅7⋅192^{3} \cdot 3 \cdot 7 \cdot 19
Sign: 11
Analytic conductor: 1.593011.59301
Root analytic conductor: 1.262141.26214
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: χ3192(797,⋅)\chi_{3192} (797, \cdot )
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 3192, ( :0), 1)(2,\ 3192,\ (\ :0),\ 1)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 3.1440431593.144043159
L(12)L(\frac12) ≈\approx 3.1440431593.144043159
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)
bad2 1−T 1 - T
3 1−T 1 - T
7 1+T 1 + T
19 1+T 1 + T
good5 (1−T)(1+T) ( 1 - T )( 1 + T )
11 1+T2 1 + T^{2}
13 (1−T)(1+T) ( 1 - T )( 1 + T )
17 1+T2 1 + T^{2}
23 (1−T)(1+T) ( 1 - T )( 1 + T )
29 (1−T)(1+T) ( 1 - T )( 1 + T )
31 1+T2 1 + T^{2}
37 1+T2 1 + T^{2}
41 (1−T)(1+T) ( 1 - T )( 1 + T )
43 (1−T)(1+T) ( 1 - T )( 1 + T )
47 1+T2 1 + T^{2}
53 (1−T)(1+T) ( 1 - T )( 1 + T )
59 (1+T)2 ( 1 + T )^{2}
61 (1+T)2 ( 1 + T )^{2}
67 1+T2 1 + T^{2}
71 (1+T)2 ( 1 + T )^{2}
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)(1+T) ( 1 - T )( 1 + T )
97 1+T2 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

−8.871871187572030857875992288122, −7.978908528074377906828360286512, −7.22591376078376709145205018755, −6.57094137252005987736383245516, −5.92682665195104732043960205839, −4.74539400389285829018088217181, −4.11995077500143505516475260619, −3.19785295686533596890691623575, −2.70454960633492071904108972221, −1.59201705226022670499623273205, 1.59201705226022670499623273205, 2.70454960633492071904108972221, 3.19785295686533596890691623575, 4.11995077500143505516475260619, 4.74539400389285829018088217181, 5.92682665195104732043960205839, 6.57094137252005987736383245516, 7.22591376078376709145205018755, 7.978908528074377906828360286512, 8.871871187572030857875992288122

Graph of the ZZ-function along the critical line