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Properties

Label 2-76e2-1.1-c1-0-63
Degree 22
Conductor 57765776
Sign −1-1
Analytic cond. 46.121546.1215
Root an. cond. 6.791286.79128
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 4·5-s − 2·9-s − 3·11-s + 2·13-s + 4·15-s + 2·17-s − 6·23-s + 11·25-s + 5·27-s − 4·29-s + 10·31-s + 3·33-s + 2·37-s − 2·39-s + 9·41-s + 4·43-s + 8·45-s + 12·47-s − 7·49-s − 2·51-s − 2·53-s + 12·55-s + 59-s − 8·61-s − 8·65-s − 9·67-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.78·5-s − 2/3·9-s − 0.904·11-s + 0.554·13-s + 1.03·15-s + 0.485·17-s − 1.25·23-s + 11/5·25-s + 0.962·27-s − 0.742·29-s + 1.79·31-s + 0.522·33-s + 0.328·37-s − 0.320·39-s + 1.40·41-s + 0.609·43-s + 1.19·45-s + 1.75·47-s − 49-s − 0.280·51-s − 0.274·53-s + 1.61·55-s + 0.130·59-s − 1.02·61-s − 0.992·65-s − 1.09·67-s + ⋯

Functional equation

Λ(s)=(5776s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 5776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(5776s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 5776 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 57765776    =    24⋅1922^{4} \cdot 19^{2}
Sign: −1-1
Analytic conductor: 46.121546.1215
Root analytic conductor: 6.791286.79128
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 5776, ( :1/2), −1)(2,\ 5776,\ (\ :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
19 1 1
good3 1+T+pT2 1 + T + p T^{2}
5 1+4T+pT2 1 + 4 T + p T^{2}
7 1+pT2 1 + p T^{2}
11 1+3T+pT2 1 + 3 T + p T^{2}
13 1−2T+pT2 1 - 2 T + p T^{2}
17 1−2T+pT2 1 - 2 T + p T^{2}
23 1+6T+pT2 1 + 6 T + p T^{2}
29 1+4T+pT2 1 + 4 T + p T^{2}
31 1−10T+pT2 1 - 10 T + p T^{2}
37 1−2T+pT2 1 - 2 T + p T^{2}
41 1−9T+pT2 1 - 9 T + p T^{2}
43 1−4T+pT2 1 - 4 T + p T^{2}
47 1−12T+pT2 1 - 12 T + p T^{2}
53 1+2T+pT2 1 + 2 T + p T^{2}
59 1−T+pT2 1 - T + p T^{2}
61 1+8T+pT2 1 + 8 T + p T^{2}
67 1+9T+pT2 1 + 9 T + p T^{2}
71 1−6T+pT2 1 - 6 T + p T^{2}
73 1+9T+pT2 1 + 9 T + p T^{2}
79 1−4T+pT2 1 - 4 T + p T^{2}
83 1−5T+pT2 1 - 5 T + p T^{2}
89 1+18T+pT2 1 + 18 T + p T^{2}
97 1−T+pT2 1 - T + p 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

−7.84291285553569302803646666567, −7.25281256893503607163488926699, −6.18133217332434823974698514600, −5.72056544906938218909518824058, −4.70361893560692538945006993689, −4.16988049041125598647554401476, −3.30209644061830425490225828028, −2.57233604129214373443319462753, −0.900922910872936613997172410594, 0, 0.900922910872936613997172410594, 2.57233604129214373443319462753, 3.30209644061830425490225828028, 4.16988049041125598647554401476, 4.70361893560692538945006993689, 5.72056544906938218909518824058, 6.18133217332434823974698514600, 7.25281256893503607163488926699, 7.84291285553569302803646666567

Graph of the ZZ-function along the critical line