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Properties

Label 2-5220-1.1-c1-0-40
Degree 22
Conductor 52205220
Sign −1-1
Analytic cond. 41.681941.6819
Root an. cond. 6.456156.45615
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s − 0.648·7-s − 3.35·11-s + 4.17·13-s − 4.82·17-s + 6.82·19-s − 5.52·23-s + 25-s + 29-s − 2.82·31-s − 0.648·35-s − 10.2·37-s − 8.17·41-s − 5.69·43-s + 2.64·47-s − 6.58·49-s + 2.87·53-s − 3.35·55-s + 13.2·59-s − 1.12·61-s + 4.17·65-s + 1.52·67-s + 8.87·71-s + 9.69·73-s + 2.17·77-s + 8.99·79-s − 1.94·83-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.244·7-s − 1.01·11-s + 1.15·13-s − 1.16·17-s + 1.56·19-s − 1.15·23-s + 0.200·25-s + 0.185·29-s − 0.506·31-s − 0.109·35-s − 1.68·37-s − 1.27·41-s − 0.868·43-s + 0.386·47-s − 0.940·49-s + 0.395·53-s − 0.451·55-s + 1.72·59-s − 0.143·61-s + 0.517·65-s + 0.186·67-s + 1.05·71-s + 1.13·73-s + 0.247·77-s + 1.01·79-s − 0.213·83-s + ⋯

Functional equation

Λ(s)=(5220s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 5220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(5220s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 5220 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 52205220    =    22⋅32⋅5⋅292^{2} \cdot 3^{2} \cdot 5 \cdot 29
Sign: −1-1
Analytic conductor: 41.681941.6819
Root analytic conductor: 6.456156.45615
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 5220, ( :1/2), −1)(2,\ 5220,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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)
bad2 1 1
3 1 1
5 1−T 1 - T
29 1−T 1 - T
good7 1+0.648T+7T2 1 + 0.648T + 7T^{2}
11 1+3.35T+11T2 1 + 3.35T + 11T^{2}
13 1−4.17T+13T2 1 - 4.17T + 13T^{2}
17 1+4.82T+17T2 1 + 4.82T + 17T^{2}
19 1−6.82T+19T2 1 - 6.82T + 19T^{2}
23 1+5.52T+23T2 1 + 5.52T + 23T^{2}
31 1+2.82T+31T2 1 + 2.82T + 31T^{2}
37 1+10.2T+37T2 1 + 10.2T + 37T^{2}
41 1+8.17T+41T2 1 + 8.17T + 41T^{2}
43 1+5.69T+43T2 1 + 5.69T + 43T^{2}
47 1−2.64T+47T2 1 - 2.64T + 47T^{2}
53 1−2.87T+53T2 1 - 2.87T + 53T^{2}
59 1−13.2T+59T2 1 - 13.2T + 59T^{2}
61 1+1.12T+61T2 1 + 1.12T + 61T^{2}
67 1−1.52T+67T2 1 - 1.52T + 67T^{2}
71 1−8.87T+71T2 1 - 8.87T + 71T^{2}
73 1−9.69T+73T2 1 - 9.69T + 73T^{2}
79 1−8.99T+79T2 1 - 8.99T + 79T^{2}
83 1+1.94T+83T2 1 + 1.94T + 83T^{2}
89 1+17.0T+89T2 1 + 17.0T + 89T^{2}
97 1+13.3T+97T2 1 + 13.3T + 97T^{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

−8.041227389211241973804789008303, −6.92687192791166052311481522380, −6.56033229582637809809321863363, −5.42716492047027328077659318231, −5.26283226594296558626517902941, −3.99316201191640441343857693944, −3.31387031175211363420609494670, −2.34865965233774095571105588383, −1.44087116678613878655112741439, 0, 1.44087116678613878655112741439, 2.34865965233774095571105588383, 3.31387031175211363420609494670, 3.99316201191640441343857693944, 5.26283226594296558626517902941, 5.42716492047027328077659318231, 6.56033229582637809809321863363, 6.92687192791166052311481522380, 8.041227389211241973804789008303

Graph of the ZZ-function along the critical line