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Properties

Label 2-342-1.1-c5-0-18
Degree 22
Conductor 342342
Sign 11
Analytic cond. 54.851254.8512
Root an. cond. 7.406167.40616
Motivic weight 55
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s + 16·4-s + 54·5-s + 104·7-s + 64·8-s + 216·10-s + 330·11-s − 46·13-s + 416·14-s + 256·16-s + 618·17-s + 361·19-s + 864·20-s + 1.32e3·22-s + 402·23-s − 209·25-s − 184·26-s + 1.66e3·28-s + 2.62e3·29-s − 2.36e3·31-s + 1.02e3·32-s + 2.47e3·34-s + 5.61e3·35-s − 1.21e4·37-s + 1.44e3·38-s + 3.45e3·40-s + 1.88e4·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.965·5-s + 0.802·7-s + 0.353·8-s + 0.683·10-s + 0.822·11-s − 0.0754·13-s + 0.567·14-s + 1/4·16-s + 0.518·17-s + 0.229·19-s + 0.482·20-s + 0.581·22-s + 0.158·23-s − 0.0668·25-s − 0.0533·26-s + 0.401·28-s + 0.580·29-s − 0.442·31-s + 0.176·32-s + 0.366·34-s + 0.774·35-s − 1.45·37-s + 0.162·38-s + 0.341·40-s + 1.75·41-s + ⋯

Functional equation

Λ(s)=(342s/2 ΓC(s) L(s)=( Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(342s/2 ΓC(s+5/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 342342    =    2⋅32⋅192 \cdot 3^{2} \cdot 19
Sign: 11
Analytic conductor: 54.851254.8512
Root analytic conductor: 7.406167.40616
Motivic weight: 55
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 342, ( :5/2), 1)(2,\ 342,\ (\ :5/2),\ 1)

Particular Values

L(3)L(3) ≈\approx 4.8768078094.876807809
L(12)L(\frac12) ≈\approx 4.8768078094.876807809
L(72)L(\frac{7}{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)
bad2 1−p2T 1 - p^{2} T
3 1 1
19 1−p2T 1 - p^{2} T
good5 1−54T+p5T2 1 - 54 T + p^{5} T^{2}
7 1−104T+p5T2 1 - 104 T + p^{5} T^{2}
11 1−30pT+p5T2 1 - 30 p T + p^{5} T^{2}
13 1+46T+p5T2 1 + 46 T + p^{5} T^{2}
17 1−618T+p5T2 1 - 618 T + p^{5} T^{2}
23 1−402T+p5T2 1 - 402 T + p^{5} T^{2}
29 1−2628T+p5T2 1 - 2628 T + p^{5} T^{2}
31 1+2368T+p5T2 1 + 2368 T + p^{5} T^{2}
37 1+12130T+p5T2 1 + 12130 T + p^{5} T^{2}
41 1−18864T+p5T2 1 - 18864 T + p^{5} T^{2}
43 1+10408T+p5T2 1 + 10408 T + p^{5} T^{2}
47 1−4770T+p5T2 1 - 4770 T + p^{5} T^{2}
53 1−19452T+p5T2 1 - 19452 T + p^{5} T^{2}
59 1+30528T+p5T2 1 + 30528 T + p^{5} T^{2}
61 1−11138T+p5T2 1 - 11138 T + p^{5} T^{2}
67 1−49508T+p5T2 1 - 49508 T + p^{5} T^{2}
71 1+7572T+p5T2 1 + 7572 T + p^{5} T^{2}
73 1−2342T+p5T2 1 - 2342 T + p^{5} T^{2}
79 1−22424T+p5T2 1 - 22424 T + p^{5} T^{2}
83 1−46734T+p5T2 1 - 46734 T + p^{5} T^{2}
89 1−70104T+p5T2 1 - 70104 T + p^{5} T^{2}
97 1−105710T+p5T2 1 - 105710 T + p^{5} 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

−10.80314954226782226125677517767, −9.859706608272543043554854684048, −8.899525852228162904215314516226, −7.71924160064499133390242220788, −6.63319921564015299299007205400, −5.69068860135008260059566301013, −4.84359547032255771859246325878, −3.61970527924208798997624406449, −2.21313862164321987933051661464, −1.22804966424100054992964679446, 1.22804966424100054992964679446, 2.21313862164321987933051661464, 3.61970527924208798997624406449, 4.84359547032255771859246325878, 5.69068860135008260059566301013, 6.63319921564015299299007205400, 7.71924160064499133390242220788, 8.899525852228162904215314516226, 9.859706608272543043554854684048, 10.80314954226782226125677517767

Graph of the ZZ-function along the critical line