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Properties

Label 2-396-1.1-c5-0-19
Degree 22
Conductor 396396
Sign −1-1
Analytic cond. 63.511963.5119
Root an. cond. 7.969447.96944
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  + 79·5-s − 50·7-s − 121·11-s − 380·13-s + 1.15e3·17-s − 1.82e3·19-s − 3.59e3·23-s + 3.11e3·25-s − 8.03e3·29-s − 2.94e3·31-s − 3.95e3·35-s + 6.97e3·37-s + 520·41-s − 2.48e3·43-s + 6.92e3·47-s − 1.43e4·49-s + 1.37e4·53-s − 9.55e3·55-s + 3.17e4·59-s + 3.41e4·61-s − 3.00e4·65-s − 6.15e4·67-s + 1.49e4·71-s − 3.64e4·73-s + 6.05e3·77-s − 2.85e4·79-s − 7.74e4·83-s + ⋯
L(s)  = 1  + 1.41·5-s − 0.385·7-s − 0.301·11-s − 0.623·13-s + 0.968·17-s − 1.15·19-s − 1.41·23-s + 0.997·25-s − 1.77·29-s − 0.550·31-s − 0.545·35-s + 0.838·37-s + 0.0483·41-s − 0.205·43-s + 0.456·47-s − 0.851·49-s + 0.670·53-s − 0.426·55-s + 1.18·59-s + 1.17·61-s − 0.881·65-s − 1.67·67-s + 0.352·71-s − 0.800·73-s + 0.116·77-s − 0.514·79-s − 1.23·83-s + ⋯

Functional equation

Λ(s)=(396s/2 ΓC(s) L(s)=(− Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 396 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}
Λ(s)=(396s/2 ΓC(s+5/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 396 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 396396    =    22⋅32⋅112^{2} \cdot 3^{2} \cdot 11
Sign: −1-1
Analytic conductor: 63.511963.5119
Root analytic conductor: 7.969447.96944
Motivic weight: 55
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 396, ( :5/2), −1)(2,\ 396,\ (\ :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
11 1+p2T 1 + p^{2} T
good5 1−79T+p5T2 1 - 79 T + p^{5} T^{2}
7 1+50T+p5T2 1 + 50 T + p^{5} T^{2}
13 1+380T+p5T2 1 + 380 T + p^{5} T^{2}
17 1−1154T+p5T2 1 - 1154 T + p^{5} T^{2}
19 1+96pT+p5T2 1 + 96 p T + p^{5} T^{2}
23 1+3591T+p5T2 1 + 3591 T + p^{5} T^{2}
29 1+8032T+p5T2 1 + 8032 T + p^{5} T^{2}
31 1+95pT+p5T2 1 + 95 p T + p^{5} T^{2}
37 1−6979T+p5T2 1 - 6979 T + p^{5} T^{2}
41 1−520T+p5T2 1 - 520 T + p^{5} T^{2}
43 1+2486T+p5T2 1 + 2486 T + p^{5} T^{2}
47 1−6920T+p5T2 1 - 6920 T + p^{5} T^{2}
53 1−13718T+p5T2 1 - 13718 T + p^{5} T^{2}
59 1−31779T+p5T2 1 - 31779 T + p^{5} T^{2}
61 1−34156T+p5T2 1 - 34156 T + p^{5} T^{2}
67 1+61503T+p5T2 1 + 61503 T + p^{5} T^{2}
71 1−14971T+p5T2 1 - 14971 T + p^{5} T^{2}
73 1+36444T+p5T2 1 + 36444 T + p^{5} T^{2}
79 1+28538T+p5T2 1 + 28538 T + p^{5} T^{2}
83 1+77482T+p5T2 1 + 77482 T + p^{5} T^{2}
89 1+36271T+p5T2 1 + 36271 T + p^{5} T^{2}
97 1+49799T+p5T2 1 + 49799 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

−9.911588091609013567115049231733, −9.400535098957028482525819640297, −8.203905151590193298589715630096, −7.10622176849326186479276478497, −5.98071167938923516000357899507, −5.46103433558983747756893268791, −4.01469984723911304191781950560, −2.57421211616639833393004829869, −1.69371768636051967108131171102, 0, 1.69371768636051967108131171102, 2.57421211616639833393004829869, 4.01469984723911304191781950560, 5.46103433558983747756893268791, 5.98071167938923516000357899507, 7.10622176849326186479276478497, 8.203905151590193298589715630096, 9.400535098957028482525819640297, 9.911588091609013567115049231733

Graph of the ZZ-function along the critical line