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Properties

Label 2-460e2-1.1-c1-0-17
Degree 22
Conductor 211600211600
Sign 11
Analytic cond. 1689.631689.63
Root an. cond. 41.105141.1051
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s + 2·7-s + 6·9-s − 13-s − 6·21-s − 9·27-s − 3·29-s − 3·31-s − 8·37-s + 3·39-s + 3·41-s + 2·43-s − 11·47-s − 3·49-s − 14·53-s + 8·59-s + 4·61-s + 12·63-s + 4·67-s − 7·71-s + 9·73-s + 9·81-s − 4·83-s + 9·87-s + 2·89-s − 2·91-s + 9·93-s + ⋯
L(s)  = 1  − 1.73·3-s + 0.755·7-s + 2·9-s − 0.277·13-s − 1.30·21-s − 1.73·27-s − 0.557·29-s − 0.538·31-s − 1.31·37-s + 0.480·39-s + 0.468·41-s + 0.304·43-s − 1.60·47-s − 3/7·49-s − 1.92·53-s + 1.04·59-s + 0.512·61-s + 1.51·63-s + 0.488·67-s − 0.830·71-s + 1.05·73-s + 81-s − 0.439·83-s + 0.964·87-s + 0.211·89-s − 0.209·91-s + 0.933·93-s + ⋯

Functional equation

Λ(s)=(211600s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 211600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(211600s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 211600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 211600211600    =    24⋅52⋅2322^{4} \cdot 5^{2} \cdot 23^{2}
Sign: 11
Analytic conductor: 1689.631689.63
Root analytic conductor: 41.105141.1051
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 211600, ( :1/2), 1)(2,\ 211600,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.89997822040.8999782204
L(12)L(\frac12) ≈\approx 0.89997822040.8999782204
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
5 1 1
23 1 1
good3 1+pT+pT2 1 + p T + p T^{2}
7 1−2T+pT2 1 - 2 T + p T^{2}
11 1+pT2 1 + p T^{2}
13 1+T+pT2 1 + T + p T^{2}
17 1+pT2 1 + p T^{2}
19 1+pT2 1 + p T^{2}
29 1+3T+pT2 1 + 3 T + p T^{2}
31 1+3T+pT2 1 + 3 T + p T^{2}
37 1+8T+pT2 1 + 8 T + p T^{2}
41 1−3T+pT2 1 - 3 T + p T^{2}
43 1−2T+pT2 1 - 2 T + p T^{2}
47 1+11T+pT2 1 + 11 T + p T^{2}
53 1+14T+pT2 1 + 14 T + p T^{2}
59 1−8T+pT2 1 - 8 T + p T^{2}
61 1−4T+pT2 1 - 4 T + p T^{2}
67 1−4T+pT2 1 - 4 T + p T^{2}
71 1+7T+pT2 1 + 7 T + p T^{2}
73 1−9T+pT2 1 - 9 T + p T^{2}
79 1+pT2 1 + p T^{2}
83 1+4T+pT2 1 + 4 T + p T^{2}
89 1−2T+pT2 1 - 2 T + p T^{2}
97 1−18T+pT2 1 - 18 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

−12.82732481424720, −12.57063843305134, −11.87656361637473, −11.57989818059086, −11.21560546156270, −10.82076245491319, −10.36973084548055, −9.804236161247565, −9.490913378415842, −8.686206991602432, −8.276320762643828, −7.503608891232912, −7.316718943552641, −6.652479663103066, −6.204574367447130, −5.747642158823935, −5.192893778009595, −4.784839244884510, −4.535699799554157, −3.652064656866985, −3.237065305591014, −2.040637519694272, −1.808367746583560, −0.9999828745508161, −0.3455532817004170, 0.3455532817004170, 0.9999828745508161, 1.808367746583560, 2.040637519694272, 3.237065305591014, 3.652064656866985, 4.535699799554157, 4.784839244884510, 5.192893778009595, 5.747642158823935, 6.204574367447130, 6.652479663103066, 7.316718943552641, 7.503608891232912, 8.276320762643828, 8.686206991602432, 9.490913378415842, 9.804236161247565, 10.36973084548055, 10.82076245491319, 11.21560546156270, 11.57989818059086, 11.87656361637473, 12.57063843305134, 12.82732481424720

Graph of the ZZ-function along the critical line