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Properties

Label 2-1058-1.1-c1-0-3
Degree 22
Conductor 10581058
Sign 11
Analytic cond. 8.448178.44817
Root an. cond. 2.906572.90657
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  − 2-s + 4-s − 4·5-s + 4·7-s − 8-s − 3·9-s + 4·10-s − 2·11-s − 2·13-s − 4·14-s + 16-s + 2·17-s + 3·18-s + 2·19-s − 4·20-s + 2·22-s + 11·25-s + 2·26-s + 4·28-s + 2·29-s − 32-s − 2·34-s − 16·35-s − 3·36-s + 4·37-s − 2·38-s + 4·40-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 1.78·5-s + 1.51·7-s − 0.353·8-s − 9-s + 1.26·10-s − 0.603·11-s − 0.554·13-s − 1.06·14-s + 1/4·16-s + 0.485·17-s + 0.707·18-s + 0.458·19-s − 0.894·20-s + 0.426·22-s + 11/5·25-s + 0.392·26-s + 0.755·28-s + 0.371·29-s − 0.176·32-s − 0.342·34-s − 2.70·35-s − 1/2·36-s + 0.657·37-s − 0.324·38-s + 0.632·40-s + ⋯

Functional equation

Λ(s)=(1058s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1058 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1058s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1058 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 10581058    =    2⋅2322 \cdot 23^{2}
Sign: 11
Analytic conductor: 8.448178.44817
Root analytic conductor: 2.906572.90657
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1058, ( :1/2), 1)(2,\ 1058,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.75833012800.7583301280
L(12)L(\frac12) ≈\approx 0.75833012800.7583301280
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+T 1 + T
23 1 1
good3 1+pT2 1 + p T^{2}
5 1+4T+pT2 1 + 4 T + p T^{2}
7 1−4T+pT2 1 - 4 T + p T^{2}
11 1+2T+pT2 1 + 2 T + p T^{2}
13 1+2T+pT2 1 + 2 T + p T^{2}
17 1−2T+pT2 1 - 2 T + p T^{2}
19 1−2T+pT2 1 - 2 T + p T^{2}
29 1−2T+pT2 1 - 2 T + p T^{2}
31 1+pT2 1 + p T^{2}
37 1−4T+pT2 1 - 4 T + p T^{2}
41 1−6T+pT2 1 - 6 T + p T^{2}
43 1+10T+pT2 1 + 10 T + p T^{2}
47 1+pT2 1 + p T^{2}
53 1−4T+pT2 1 - 4 T + p T^{2}
59 1−12T+pT2 1 - 12 T + p T^{2}
61 1−8T+pT2 1 - 8 T + p T^{2}
67 1−10T+pT2 1 - 10 T + p T^{2}
71 1+pT2 1 + p T^{2}
73 1−6T+pT2 1 - 6 T + p T^{2}
79 1−12T+pT2 1 - 12 T + p T^{2}
83 1+14T+pT2 1 + 14 T + p T^{2}
89 1−6T+pT2 1 - 6 T + p T^{2}
97 1+6T+pT2 1 + 6 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

−9.947689550336840395250376815301, −8.683650556938610005541061706290, −8.159724093271066632099122061158, −7.76953499301176804540241574451, −6.97395956350401904764981202712, −5.45714573817728122372138066179, −4.72575079143831508702325515450, −3.56810168580414870732367804186, −2.44935456312664529103463287959, −0.73237197917342842336436187115, 0.73237197917342842336436187115, 2.44935456312664529103463287959, 3.56810168580414870732367804186, 4.72575079143831508702325515450, 5.45714573817728122372138066179, 6.97395956350401904764981202712, 7.76953499301176804540241574451, 8.159724093271066632099122061158, 8.683650556938610005541061706290, 9.947689550336840395250376815301

Graph of the ZZ-function along the critical line