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Properties

Label 2-1160-1.1-c1-0-20
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  + 2.36·3-s + 5-s + 2.16·7-s + 2.58·9-s + 1.75·11-s + 2.60·13-s + 2.36·15-s − 5.49·17-s + 5.90·19-s + 5.12·21-s − 7.29·23-s + 25-s − 0.978·27-s − 29-s − 0.169·31-s + 4.15·33-s + 2.16·35-s + 3.75·37-s + 6.16·39-s − 10.3·41-s + 4.55·43-s + 2.58·45-s + 1.17·47-s − 2.29·49-s − 12.9·51-s + 11.8·53-s + 1.75·55-s + ⋯
L(s)  = 1  + 1.36·3-s + 0.447·5-s + 0.820·7-s + 0.862·9-s + 0.529·11-s + 0.723·13-s + 0.610·15-s − 1.33·17-s + 1.35·19-s + 1.11·21-s − 1.52·23-s + 0.200·25-s − 0.188·27-s − 0.185·29-s − 0.0305·31-s + 0.722·33-s + 0.366·35-s + 0.617·37-s + 0.986·39-s − 1.62·41-s + 0.694·43-s + 0.385·45-s + 0.171·47-s − 0.327·49-s − 1.81·51-s + 1.62·53-s + 0.236·55-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 3.0985493753.098549375
L(12)L(\frac12) ≈\approx 3.0985493753.098549375
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 1−2.36T+3T2 1 - 2.36T + 3T^{2}
7 1−2.16T+7T2 1 - 2.16T + 7T^{2}
11 1−1.75T+11T2 1 - 1.75T + 11T^{2}
13 1−2.60T+13T2 1 - 2.60T + 13T^{2}
17 1+5.49T+17T2 1 + 5.49T + 17T^{2}
19 1−5.90T+19T2 1 - 5.90T + 19T^{2}
23 1+7.29T+23T2 1 + 7.29T + 23T^{2}
31 1+0.169T+31T2 1 + 0.169T + 31T^{2}
37 1−3.75T+37T2 1 - 3.75T + 37T^{2}
41 1+10.3T+41T2 1 + 10.3T + 41T^{2}
43 1−4.55T+43T2 1 - 4.55T + 43T^{2}
47 1−1.17T+47T2 1 - 1.17T + 47T^{2}
53 1−11.8T+53T2 1 - 11.8T + 53T^{2}
59 1+4.11T+59T2 1 + 4.11T + 59T^{2}
61 1−1.00T+61T2 1 - 1.00T + 61T^{2}
67 1+8.48T+67T2 1 + 8.48T + 67T^{2}
71 1−6.10T+71T2 1 - 6.10T + 71T^{2}
73 1+1.72T+73T2 1 + 1.72T + 73T^{2}
79 1+2.02T+79T2 1 + 2.02T + 79T^{2}
83 1−1.86T+83T2 1 - 1.86T + 83T^{2}
89 1−16.5T+89T2 1 - 16.5T + 89T^{2}
97 1−0.464T+97T2 1 - 0.464T + 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

−9.526575145113876665310531888466, −8.930590367603212666255588462952, −8.266399900780545732894954577342, −7.57316114623749431934838096163, −6.54995551773627559645820167044, −5.53786481669365680781751263131, −4.36110763073305416691484035441, −3.54343183443131347105662667459, −2.38600569989539124294696129828, −1.53198318254994865028898667172, 1.53198318254994865028898667172, 2.38600569989539124294696129828, 3.54343183443131347105662667459, 4.36110763073305416691484035441, 5.53786481669365680781751263131, 6.54995551773627559645820167044, 7.57316114623749431934838096163, 8.266399900780545732894954577342, 8.930590367603212666255588462952, 9.526575145113876665310531888466

Graph of the ZZ-function along the critical line