↑

Properties

Label 2-1160-1.1-c1-0-21
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3.04·3-s + 5-s + 3.43·7-s + 6.26·9-s − 0.254·11-s − 2.66·13-s + 3.04·15-s + 0.449·17-s − 4.69·19-s + 10.4·21-s − 0.415·23-s + 25-s + 9.93·27-s + 29-s − 5.51·31-s − 0.774·33-s + 3.43·35-s − 7.29·37-s − 8.12·39-s − 4.86·41-s + 2.44·43-s + 6.26·45-s − 12.6·47-s + 4.77·49-s + 1.36·51-s + 8.60·53-s − 0.254·55-s + ⋯
L(s)  = 1  + 1.75·3-s + 0.447·5-s + 1.29·7-s + 2.08·9-s − 0.0767·11-s − 0.740·13-s + 0.785·15-s + 0.109·17-s − 1.07·19-s + 2.27·21-s − 0.0865·23-s + 0.200·25-s + 1.91·27-s + 0.185·29-s − 0.991·31-s − 0.134·33-s + 0.579·35-s − 1.19·37-s − 1.30·39-s − 0.759·41-s + 0.373·43-s + 0.933·45-s − 1.84·47-s + 0.681·49-s + 0.191·51-s + 1.18·53-s − 0.0343·55-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 3.5168754283.516875428
L(12)L(\frac12) ≈\approx 3.5168754283.516875428
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 1−3.04T+3T2 1 - 3.04T + 3T^{2}
7 1−3.43T+7T2 1 - 3.43T + 7T^{2}
11 1+0.254T+11T2 1 + 0.254T + 11T^{2}
13 1+2.66T+13T2 1 + 2.66T + 13T^{2}
17 1−0.449T+17T2 1 - 0.449T + 17T^{2}
19 1+4.69T+19T2 1 + 4.69T + 19T^{2}
23 1+0.415T+23T2 1 + 0.415T + 23T^{2}
31 1+5.51T+31T2 1 + 5.51T + 31T^{2}
37 1+7.29T+37T2 1 + 7.29T + 37T^{2}
41 1+4.86T+41T2 1 + 4.86T + 41T^{2}
43 1−2.44T+43T2 1 - 2.44T + 43T^{2}
47 1+12.6T+47T2 1 + 12.6T + 47T^{2}
53 1−8.60T+53T2 1 - 8.60T + 53T^{2}
59 1−4.16T+59T2 1 - 4.16T + 59T^{2}
61 1−1.77T+61T2 1 - 1.77T + 61T^{2}
67 1−11.1T+67T2 1 - 11.1T + 67T^{2}
71 1+12.1T+71T2 1 + 12.1T + 71T^{2}
73 1−4.11T+73T2 1 - 4.11T + 73T^{2}
79 1−11.7T+79T2 1 - 11.7T + 79T^{2}
83 1+0.185T+83T2 1 + 0.185T + 83T^{2}
89 1+12.9T+89T2 1 + 12.9T + 89T^{2}
97 1−9.24T+97T2 1 - 9.24T + 97T^{2}
show more
show less
   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

−9.681295010611156202041081217687, −8.722020755307850858941598180697, −8.356329709706303822011415875665, −7.54633122895044298946965410130, −6.78256103649841922287883016233, −5.30212568897666243472800531294, −4.47354503152814597240065825288, −3.46874519434779586050589746420, −2.29452224919892585739625483141, −1.71989866536344280379084609593, 1.71989866536344280379084609593, 2.29452224919892585739625483141, 3.46874519434779586050589746420, 4.47354503152814597240065825288, 5.30212568897666243472800531294, 6.78256103649841922287883016233, 7.54633122895044298946965410130, 8.356329709706303822011415875665, 8.722020755307850858941598180697, 9.681295010611156202041081217687

Graph of the ZZ-function along the critical line