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Properties

Label 2-1160-1160.579-c0-0-3
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 − 1.61·3-s + 4-s − 5-s − 1.61·6-s − 0.618·7-s + 8-s + 1.61·9-s − 10-s − 1.61·12-s + 1.61·13-s − 0.618·14-s + 1.61·15-s + 16-s + 0.618·17-s + 1.61·18-s − 20-s + 1.00·21-s + 1.61·23-s − 1.61·24-s + 25-s + 1.61·26-s − 27-s − 0.618·28-s − 29-s + 1.61·30-s − 0.618·31-s + ⋯
L(s)  = 1  + 2-s − 1.61·3-s + 4-s − 5-s − 1.61·6-s − 0.618·7-s + 8-s + 1.61·9-s − 10-s − 1.61·12-s + 1.61·13-s − 0.618·14-s + 1.61·15-s + 16-s + 0.618·17-s + 1.61·18-s − 20-s + 1.00·21-s + 1.61·23-s − 1.61·24-s + 25-s + 1.61·26-s − 27-s − 0.618·28-s − 29-s + 1.61·30-s − 0.618·31-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 1.0501047621.050104762
L(12)L(\frac12) ≈\approx 1.0501047621.050104762
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+1.61T+T2 1 + 1.61T + T^{2}
7 1+0.618T+T2 1 + 0.618T + T^{2}
11 1−T2 1 - T^{2}
13 1−1.61T+T2 1 - 1.61T + T^{2}
17 1−0.618T+T2 1 - 0.618T + T^{2}
19 1−T2 1 - T^{2}
23 1−1.61T+T2 1 - 1.61T + T^{2}
31 1+0.618T+T2 1 + 0.618T + T^{2}
37 1−T2 1 - T^{2}
41 1−T2 1 - T^{2}
43 1−0.618T+T2 1 - 0.618T + T^{2}
47 1−T2 1 - T^{2}
53 1+0.618T+T2 1 + 0.618T + T^{2}
59 1+1.61T+T2 1 + 1.61T + T^{2}
61 1−1.61T+T2 1 - 1.61T + T^{2}
67 1−T2 1 - T^{2}
71 1−T2 1 - T^{2}
73 1−0.618T+T2 1 - 0.618T + T^{2}
79 1−1.61T+T2 1 - 1.61T + T^{2}
83 1−T2 1 - T^{2}
89 1−T2 1 - T^{2}
97 1+1.61T+T2 1 + 1.61T + 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

−10.61538254760137338434849738194, −9.319628248660324443625541919409, −8.026497732542470937651210227160, −7.07178375486281940218648856737, −6.48277511298224935502289049105, −5.69465854576682597227114274798, −4.97309202629132727401051446891, −3.95095855178449911771354271405, −3.25267216476052250460876334024, −1.15959764657448848261523418261, 1.15959764657448848261523418261, 3.25267216476052250460876334024, 3.95095855178449911771354271405, 4.97309202629132727401051446891, 5.69465854576682597227114274798, 6.48277511298224935502289049105, 7.07178375486281940218648856737, 8.026497732542470937651210227160, 9.319628248660324443625541919409, 10.61538254760137338434849738194

Graph of the ZZ-function along the critical line