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Properties

Label 2-459-51.50-c0-0-1
Degree 22
Conductor 459459
Sign 11
Analytic cond. 0.2290700.229070
Root an. cond. 0.4786130.478613
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  + 4-s + 5-s − 2·11-s − 13-s + 16-s − 17-s − 19-s + 20-s + 23-s + 29-s + 41-s − 43-s − 2·44-s + 49-s − 52-s − 2·55-s + 64-s − 65-s − 67-s − 68-s + 71-s − 76-s + 80-s − 85-s + 92-s − 95-s − 103-s + ⋯
L(s)  = 1  + 4-s + 5-s − 2·11-s − 13-s + 16-s − 17-s − 19-s + 20-s + 23-s + 29-s + 41-s − 43-s − 2·44-s + 49-s − 52-s − 2·55-s + 64-s − 65-s − 67-s − 68-s + 71-s − 76-s + 80-s − 85-s + 92-s − 95-s − 103-s + ⋯

Functional equation

Λ(s)=(459s/2 ΓC(s) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 459 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
Λ(s)=(459s/2 ΓC(s) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 459 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 459459    =    33⋅173^{3} \cdot 17
Sign: 11
Analytic conductor: 0.2290700.229070
Root analytic conductor: 0.4786130.478613
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: χ459(458,⋅)\chi_{459} (458, \cdot )
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 459, ( :0), 1)(2,\ 459,\ (\ :0),\ 1)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.0745056841.074505684
L(12)L(\frac12) ≈\approx 1.0745056841.074505684
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 1
17 1+T 1 + T
good2 (1−T)(1+T) ( 1 - T )( 1 + T )
5 1−T+T2 1 - T + T^{2}
7 (1−T)(1+T) ( 1 - T )( 1 + T )
11 (1+T)2 ( 1 + T )^{2}
13 1+T+T2 1 + T + T^{2}
19 1+T+T2 1 + T + T^{2}
23 1−T+T2 1 - T + T^{2}
29 1−T+T2 1 - T + T^{2}
31 (1−T)(1+T) ( 1 - T )( 1 + T )
37 (1−T)(1+T) ( 1 - T )( 1 + T )
41 1−T+T2 1 - T + T^{2}
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)(1+T) ( 1 - T )( 1 + T )
67 1+T+T2 1 + T + T^{2}
71 1−T+T2 1 - T + 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−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

−10.98332821954406819926729736269, −10.49809225540244982376634530742, −9.743262152239776875460018324323, −8.496183220853762622292772169540, −7.51061952797034652984045972716, −6.64264709677162462110086688267, −5.67142322826064861440926168592, −4.78240329404948759354773029217, −2.74916842416376406048180467989, −2.19015086381984588647439184289, 2.19015086381984588647439184289, 2.74916842416376406048180467989, 4.78240329404948759354773029217, 5.67142322826064861440926168592, 6.64264709677162462110086688267, 7.51061952797034652984045972716, 8.496183220853762622292772169540, 9.743262152239776875460018324323, 10.49809225540244982376634530742, 10.98332821954406819926729736269

Graph of the ZZ-function along the critical line