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Properties

Label 2-1160-1160.579-c0-0-7
Degree 22
Conductor 11601160
Sign 11
Analytic cond. 0.5789150.578915
Root an. cond. 0.7608640.760864
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 0.618·3-s + 4-s − 5-s + 0.618·6-s + 1.61·7-s + 8-s − 0.618·9-s − 10-s + 0.618·12-s − 0.618·13-s + 1.61·14-s − 0.618·15-s + 16-s − 1.61·17-s − 0.618·18-s − 20-s + 1.00·21-s − 0.618·23-s + 0.618·24-s + 25-s − 0.618·26-s − 27-s + 1.61·28-s − 29-s − 0.618·30-s + 1.61·31-s + ⋯
L(s)  = 1  + 2-s + 0.618·3-s + 4-s − 5-s + 0.618·6-s + 1.61·7-s + 8-s − 0.618·9-s − 10-s + 0.618·12-s − 0.618·13-s + 1.61·14-s − 0.618·15-s + 16-s − 1.61·17-s − 0.618·18-s − 20-s + 1.00·21-s − 0.618·23-s + 0.618·24-s + 25-s − 0.618·26-s − 27-s + 1.61·28-s − 29-s − 0.618·30-s + 1.61·31-s + ⋯

Functional equation

Λ(s)=(1160s/2 ΓC(s) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
Λ(s)=(1160s/2 ΓC(s) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 11601160    =    23⋅5⋅292^{3} \cdot 5 \cdot 29
Sign: 11
Analytic conductor: 0.5789150.578915
Root analytic conductor: 0.7608640.760864
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1160(579,⋅)\chi_{1160} (579, \cdot )
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1160, ( :0), 1)(2,\ 1160,\ (\ :0),\ 1)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 2.1267847612.126784761
L(12)L(\frac12) ≈\approx 2.1267847612.126784761
L(1)L(1) 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
5 1+T 1 + T
29 1+T 1 + T
good3 1−0.618T+T2 1 - 0.618T + T^{2}
7 1−1.61T+T2 1 - 1.61T + T^{2}
11 1−T2 1 - T^{2}
13 1+0.618T+T2 1 + 0.618T + T^{2}
17 1+1.61T+T2 1 + 1.61T + T^{2}
19 1−T2 1 - T^{2}
23 1+0.618T+T2 1 + 0.618T + T^{2}
31 1−1.61T+T2 1 - 1.61T + T^{2}
37 1−T2 1 - T^{2}
41 1−T2 1 - T^{2}
43 1+1.61T+T2 1 + 1.61T + T^{2}
47 1−T2 1 - T^{2}
53 1−1.61T+T2 1 - 1.61T + T^{2}
59 1−0.618T+T2 1 - 0.618T + T^{2}
61 1+0.618T+T2 1 + 0.618T + T^{2}
67 1−T2 1 - T^{2}
71 1−T2 1 - T^{2}
73 1+1.61T+T2 1 + 1.61T + T^{2}
79 1+0.618T+T2 1 + 0.618T + T^{2}
83 1−T2 1 - T^{2}
89 1−T2 1 - T^{2}
97 1−0.618T+T2 1 - 0.618T + 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

−10.25297148081094490810686177902, −8.787478978343084718482855872707, −8.221363363011877229832579870045, −7.58744122980974681960875477848, −6.73584101265995587748568802544, −5.48041984461398223489674192397, −4.61487544549217139940640306946, −4.05929876473963900584595126359, −2.82884429279692423108327934738, −1.92480588808352345743983604656, 1.92480588808352345743983604656, 2.82884429279692423108327934738, 4.05929876473963900584595126359, 4.61487544549217139940640306946, 5.48041984461398223489674192397, 6.73584101265995587748568802544, 7.58744122980974681960875477848, 8.221363363011877229832579870045, 8.787478978343084718482855872707, 10.25297148081094490810686177902

Graph of the ZZ-function along the critical line