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Properties

Label 2-459-1.1-c1-0-17
Degree 22
Conductor 459459
Sign 11
Analytic cond. 3.665133.66513
Root an. cond. 1.914451.91445
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 2·4-s + 4·5-s + 7-s + 8·10-s − 6·11-s + 13-s + 2·14-s − 4·16-s + 17-s − 7·19-s + 8·20-s − 12·22-s + 4·23-s + 11·25-s + 2·26-s + 2·28-s + 6·29-s − 8·31-s − 8·32-s + 2·34-s + 4·35-s + 37-s − 14·38-s + 4·43-s − 12·44-s + 8·46-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s + 1.78·5-s + 0.377·7-s + 2.52·10-s − 1.80·11-s + 0.277·13-s + 0.534·14-s − 16-s + 0.242·17-s − 1.60·19-s + 1.78·20-s − 2.55·22-s + 0.834·23-s + 11/5·25-s + 0.392·26-s + 0.377·28-s + 1.11·29-s − 1.43·31-s − 1.41·32-s + 0.342·34-s + 0.676·35-s + 0.164·37-s − 2.27·38-s + 0.609·43-s − 1.80·44-s + 1.17·46-s + ⋯

Functional equation

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

Invariants

Degree: 22
Conductor: 459459    =    33⋅173^{3} \cdot 17
Sign: 11
Analytic conductor: 3.665133.66513
Root analytic conductor: 1.914451.91445
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 459, ( :1/2), 1)(2,\ 459,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 3.3644606323.364460632
L(12)L(\frac12) ≈\approx 3.3644606323.364460632
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)
bad3 1 1
17 1−T 1 - T
good2 1−pT+pT2 1 - p T + p T^{2}
5 1−4T+pT2 1 - 4 T + p T^{2}
7 1−T+pT2 1 - T + p T^{2}
11 1+6T+pT2 1 + 6 T + p T^{2}
13 1−T+pT2 1 - T + p T^{2}
19 1+7T+pT2 1 + 7 T + p T^{2}
23 1−4T+pT2 1 - 4 T + p T^{2}
29 1−6T+pT2 1 - 6 T + p T^{2}
31 1+8T+pT2 1 + 8 T + p T^{2}
37 1−T+pT2 1 - T + p T^{2}
41 1+pT2 1 + p T^{2}
43 1−4T+pT2 1 - 4 T + p T^{2}
47 1−6T+pT2 1 - 6 T + p T^{2}
53 1+pT2 1 + p T^{2}
59 1−6T+pT2 1 - 6 T + p T^{2}
61 1+7T+pT2 1 + 7 T + p T^{2}
67 1−T+pT2 1 - T + p T^{2}
71 1+4T+pT2 1 + 4 T + p T^{2}
73 1−3T+pT2 1 - 3 T + p T^{2}
79 1+9T+pT2 1 + 9 T + p T^{2}
83 1−14T+pT2 1 - 14 T + p T^{2}
89 1+14T+pT2 1 + 14 T + p T^{2}
97 1+T+pT2 1 + T + p 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

−10.90508481410394760093183518479, −10.51882698778784990220643195703, −9.355188181832179362346103161628, −8.379898543961470967191694971495, −6.92940427570655812640731061069, −5.93408556270929922136180838803, −5.37713357577775960919289343412, −4.56441091173163015164546337868, −2.90936379528655043489992111903, −2.09364906131822284439528208110, 2.09364906131822284439528208110, 2.90936379528655043489992111903, 4.56441091173163015164546337868, 5.37713357577775960919289343412, 5.93408556270929922136180838803, 6.92940427570655812640731061069, 8.379898543961470967191694971495, 9.355188181832179362346103161628, 10.51882698778784990220643195703, 10.90508481410394760093183518479

Graph of the ZZ-function along the critical line