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Properties

Label 2-1160-1.1-c1-0-10
Degree 22
Conductor 11601160
Sign 11
Analytic cond. 9.262649.26264
Root an. cond. 3.043453.04345
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.857·3-s + 5-s + 2.23·7-s − 2.26·9-s − 3.21·11-s + 3.25·13-s + 0.857·15-s + 0.870·17-s + 5.02·19-s + 1.91·21-s + 8.46·23-s + 25-s − 4.51·27-s + 29-s + 0.0467·31-s − 2.75·33-s + 2.23·35-s + 11.1·37-s + 2.78·39-s − 2.47·41-s + 2.87·43-s − 2.26·45-s + 2.20·47-s − 1.99·49-s + 0.746·51-s − 11.7·53-s − 3.21·55-s + ⋯
L(s)  = 1  + 0.495·3-s + 0.447·5-s + 0.845·7-s − 0.754·9-s − 0.970·11-s + 0.901·13-s + 0.221·15-s + 0.211·17-s + 1.15·19-s + 0.418·21-s + 1.76·23-s + 0.200·25-s − 0.869·27-s + 0.185·29-s + 0.00839·31-s − 0.480·33-s + 0.378·35-s + 1.82·37-s + 0.446·39-s − 0.386·41-s + 0.437·43-s − 0.337·45-s + 0.321·47-s − 0.284·49-s + 0.104·51-s − 1.61·53-s − 0.433·55-s + ⋯

Functional equation

Λ(s)=(1160s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1160s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, 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: 9.262649.26264
Root analytic conductor: 3.043453.04345
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1160, ( :1/2), 1)(2,\ 1160,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.2408345192.240834519
L(12)L(\frac12) ≈\approx 2.2408345192.240834519
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−T 1 - T
29 1−T 1 - T
good3 1−0.857T+3T2 1 - 0.857T + 3T^{2}
7 1−2.23T+7T2 1 - 2.23T + 7T^{2}
11 1+3.21T+11T2 1 + 3.21T + 11T^{2}
13 1−3.25T+13T2 1 - 3.25T + 13T^{2}
17 1−0.870T+17T2 1 - 0.870T + 17T^{2}
19 1−5.02T+19T2 1 - 5.02T + 19T^{2}
23 1−8.46T+23T2 1 - 8.46T + 23T^{2}
31 1−0.0467T+31T2 1 - 0.0467T + 31T^{2}
37 1−11.1T+37T2 1 - 11.1T + 37T^{2}
41 1+2.47T+41T2 1 + 2.47T + 41T^{2}
43 1−2.87T+43T2 1 - 2.87T + 43T^{2}
47 1−2.20T+47T2 1 - 2.20T + 47T^{2}
53 1+11.7T+53T2 1 + 11.7T + 53T^{2}
59 1+7.68T+59T2 1 + 7.68T + 59T^{2}
61 1+4.99T+61T2 1 + 4.99T + 61T^{2}
67 1−11.6T+67T2 1 - 11.6T + 67T^{2}
71 1+3.43T+71T2 1 + 3.43T + 71T^{2}
73 1+10.1T+73T2 1 + 10.1T + 73T^{2}
79 1−6.44T+79T2 1 - 6.44T + 79T^{2}
83 1−15.4T+83T2 1 - 15.4T + 83T^{2}
89 1−16.7T+89T2 1 - 16.7T + 89T^{2}
97 1+7.16T+97T2 1 + 7.16T + 97T^{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

−9.573806472521627510606059436679, −9.004091646518504259228457657671, −8.064906799861723339715545867817, −7.64675011836398438511222909282, −6.35618545730569317852916747172, −5.44735526931506380588356651845, −4.79239369429581937718802443433, −3.33150449249403906251612336501, −2.58334215227572763483064408962, −1.21399533910978966136811152060, 1.21399533910978966136811152060, 2.58334215227572763483064408962, 3.33150449249403906251612336501, 4.79239369429581937718802443433, 5.44735526931506380588356651845, 6.35618545730569317852916747172, 7.64675011836398438511222909282, 8.064906799861723339715545867817, 9.004091646518504259228457657671, 9.573806472521627510606059436679

Graph of the ZZ-function along the critical line