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Properties

Label 2-252-1.1-c5-0-9
Degree 22
Conductor 252252
Sign −1-1
Analytic cond. 40.416740.4167
Root an. cond. 6.357416.35741
Motivic weight 55
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 34·5-s − 49·7-s + 332·11-s − 1.02e3·13-s − 922·17-s + 452·19-s + 3.77e3·23-s − 1.96e3·25-s − 1.16e3·29-s − 9.79e3·31-s − 1.66e3·35-s + 2.39e3·37-s + 7.23e3·41-s + 4.65e3·43-s − 2.46e4·47-s + 2.40e3·49-s − 1.11e3·53-s + 1.12e4·55-s − 4.68e4·59-s − 9.76e3·61-s − 3.48e4·65-s − 2.62e4·67-s − 6.54e4·71-s − 5.60e3·73-s − 1.62e4·77-s − 9.84e3·79-s − 6.11e4·83-s + ⋯
L(s)  = 1  + 0.608·5-s − 0.377·7-s + 0.827·11-s − 1.68·13-s − 0.773·17-s + 0.287·19-s + 1.48·23-s − 0.630·25-s − 0.257·29-s − 1.83·31-s − 0.229·35-s + 0.287·37-s + 0.671·41-s + 0.383·43-s − 1.62·47-s + 1/7·49-s − 0.0542·53-s + 0.503·55-s − 1.75·59-s − 0.335·61-s − 1.02·65-s − 0.714·67-s − 1.54·71-s − 0.123·73-s − 0.312·77-s − 0.177·79-s − 0.973·83-s + ⋯

Functional equation

Λ(s)=(252s/2 ΓC(s) L(s)=(− Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}
Λ(s)=(252s/2 ΓC(s+5/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 252252    =    22⋅32⋅72^{2} \cdot 3^{2} \cdot 7
Sign: −1-1
Analytic conductor: 40.416740.4167
Root analytic conductor: 6.357416.35741
Motivic weight: 55
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 252, ( :5/2), −1)(2,\ 252,\ (\ :5/2),\ -1)

Particular Values

L(3)L(3) == 00
L(12)L(\frac12) == 00
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 1
7 1+p2T 1 + p^{2} T
good5 1−34T+p5T2 1 - 34 T + p^{5} T^{2}
11 1−332T+p5T2 1 - 332 T + p^{5} T^{2}
13 1+1026T+p5T2 1 + 1026 T + p^{5} T^{2}
17 1+922T+p5T2 1 + 922 T + p^{5} T^{2}
19 1−452T+p5T2 1 - 452 T + p^{5} T^{2}
23 1−3776T+p5T2 1 - 3776 T + p^{5} T^{2}
29 1+1166T+p5T2 1 + 1166 T + p^{5} T^{2}
31 1+9792T+p5T2 1 + 9792 T + p^{5} T^{2}
37 1−2390T+p5T2 1 - 2390 T + p^{5} T^{2}
41 1−7230T+p5T2 1 - 7230 T + p^{5} T^{2}
43 1−4652T+p5T2 1 - 4652 T + p^{5} T^{2}
47 1+24672T+p5T2 1 + 24672 T + p^{5} T^{2}
53 1+1110T+p5T2 1 + 1110 T + p^{5} T^{2}
59 1+46892T+p5T2 1 + 46892 T + p^{5} T^{2}
61 1+9762T+p5T2 1 + 9762 T + p^{5} T^{2}
67 1+26252T+p5T2 1 + 26252 T + p^{5} T^{2}
71 1+65440T+p5T2 1 + 65440 T + p^{5} T^{2}
73 1+5606T+p5T2 1 + 5606 T + p^{5} T^{2}
79 1+9840T+p5T2 1 + 9840 T + p^{5} T^{2}
83 1+61108T+p5T2 1 + 61108 T + p^{5} T^{2}
89 1−62958T+p5T2 1 - 62958 T + p^{5} T^{2}
97 1+37838T+p5T2 1 + 37838 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

−10.67801095880390968041019569664, −9.471608992046821572195457984735, −9.191505614547349917516899781348, −7.55006272164724197498993845134, −6.73298255321240528839049784307, −5.57970983211582409977470066376, −4.46292149411776776270220221384, −2.97761298034663055434348945471, −1.72011968859156978245597450097, 0, 1.72011968859156978245597450097, 2.97761298034663055434348945471, 4.46292149411776776270220221384, 5.57970983211582409977470066376, 6.73298255321240528839049784307, 7.55006272164724197498993845134, 9.191505614547349917516899781348, 9.471608992046821572195457984735, 10.67801095880390968041019569664

Graph of the ZZ-function along the critical line