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Properties

Label 2-336-1.1-c5-0-6
Degree 22
Conductor 336336
Sign 11
Analytic cond. 53.888953.8889
Root an. cond. 7.340917.34091
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  − 9·3-s + 24·5-s − 49·7-s + 81·9-s − 66·11-s + 98·13-s − 216·15-s − 216·17-s + 340·19-s + 441·21-s + 1.03e3·23-s − 2.54e3·25-s − 729·27-s − 2.49e3·29-s + 7.04e3·31-s + 594·33-s − 1.17e3·35-s − 1.22e4·37-s − 882·39-s + 6.46e3·41-s + 1.54e4·43-s + 1.94e3·45-s − 2.06e4·47-s + 2.40e3·49-s + 1.94e3·51-s + 3.24e4·53-s − 1.58e3·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.429·5-s − 0.377·7-s + 1/3·9-s − 0.164·11-s + 0.160·13-s − 0.247·15-s − 0.181·17-s + 0.216·19-s + 0.218·21-s + 0.409·23-s − 0.815·25-s − 0.192·27-s − 0.549·29-s + 1.31·31-s + 0.0949·33-s − 0.162·35-s − 1.46·37-s − 0.0928·39-s + 0.600·41-s + 1.27·43-s + 0.143·45-s − 1.36·47-s + 1/7·49-s + 0.104·51-s + 1.58·53-s − 0.0706·55-s + ⋯

Functional equation

Λ(s)=(336s/2 ΓC(s) L(s)=( Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(336s/2 ΓC(s+5/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 336336    =    24⋅3⋅72^{4} \cdot 3 \cdot 7
Sign: 11
Analytic conductor: 53.888953.8889
Root analytic conductor: 7.340917.34091
Motivic weight: 55
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 336, ( :5/2), 1)(2,\ 336,\ (\ :5/2),\ 1)

Particular Values

L(3)L(3) ≈\approx 1.5705941181.570594118
L(12)L(\frac12) ≈\approx 1.5705941181.570594118
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 1
3 1+p2T 1 + p^{2} T
7 1+p2T 1 + p^{2} T
good5 1−24T+p5T2 1 - 24 T + p^{5} T^{2}
11 1+6pT+p5T2 1 + 6 p T + p^{5} T^{2}
13 1−98T+p5T2 1 - 98 T + p^{5} T^{2}
17 1+216T+p5T2 1 + 216 T + p^{5} T^{2}
19 1−340T+p5T2 1 - 340 T + p^{5} T^{2}
23 1−1038T+p5T2 1 - 1038 T + p^{5} T^{2}
29 1+2490T+p5T2 1 + 2490 T + p^{5} T^{2}
31 1−7048T+p5T2 1 - 7048 T + p^{5} T^{2}
37 1+12238T+p5T2 1 + 12238 T + p^{5} T^{2}
41 1−6468T+p5T2 1 - 6468 T + p^{5} T^{2}
43 1−15412T+p5T2 1 - 15412 T + p^{5} T^{2}
47 1+20604T+p5T2 1 + 20604 T + p^{5} T^{2}
53 1−32490T+p5T2 1 - 32490 T + p^{5} T^{2}
59 1+34224T+p5T2 1 + 34224 T + p^{5} T^{2}
61 1−35654T+p5T2 1 - 35654 T + p^{5} T^{2}
67 1+12680T+p5T2 1 + 12680 T + p^{5} T^{2}
71 1−42642T+p5T2 1 - 42642 T + p^{5} T^{2}
73 1−33734T+p5T2 1 - 33734 T + p^{5} T^{2}
79 1−85108T+p5T2 1 - 85108 T + p^{5} T^{2}
83 1−106764T+p5T2 1 - 106764 T + p^{5} T^{2}
89 1−34884T+p5T2 1 - 34884 T + p^{5} T^{2}
97 1−18662T+p5T2 1 - 18662 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

−10.68713333288016604261300323906, −9.888918160455992619229168651973, −9.012313614712693100519160442273, −7.79356508267304610060163563719, −6.71098295210445021640023294193, −5.87127380338154036328048077131, −4.88927677860030034183450172715, −3.58411228101088867263840724030, −2.14830011079263387571484286844, −0.70505553769812684164230064736, 0.70505553769812684164230064736, 2.14830011079263387571484286844, 3.58411228101088867263840724030, 4.88927677860030034183450172715, 5.87127380338154036328048077131, 6.71098295210445021640023294193, 7.79356508267304610060163563719, 9.012313614712693100519160442273, 9.888918160455992619229168651973, 10.68713333288016604261300323906

Graph of the ZZ-function along the critical line