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Properties

Label 2-4000-1.1-c1-0-22
Degree 22
Conductor 40004000
Sign 11
Analytic cond. 31.940131.9401
Root an. cond. 5.651565.65156
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  − 2.71·3-s − 0.0156·7-s + 4.35·9-s + 6.41·11-s − 2.76·13-s − 2.07·17-s + 4.25·19-s + 0.0424·21-s + 5.97·23-s − 3.67·27-s + 5.37·29-s − 3.59·31-s − 17.3·33-s + 9.04·37-s + 7.48·39-s − 7.03·41-s − 4.96·43-s + 5.73·47-s − 6.99·49-s + 5.63·51-s − 10.5·53-s − 11.5·57-s + 1.13·59-s + 8.76·61-s − 0.0682·63-s + 11.0·67-s − 16.2·69-s + ⋯
L(s)  = 1  − 1.56·3-s − 0.00591·7-s + 1.45·9-s + 1.93·11-s − 0.765·13-s − 0.503·17-s + 0.976·19-s + 0.00926·21-s + 1.24·23-s − 0.707·27-s + 0.998·29-s − 0.645·31-s − 3.02·33-s + 1.48·37-s + 1.19·39-s − 1.09·41-s − 0.756·43-s + 0.836·47-s − 0.999·49-s + 0.788·51-s − 1.45·53-s − 1.52·57-s + 0.147·59-s + 1.12·61-s − 0.00859·63-s + 1.34·67-s − 1.95·69-s + ⋯

Functional equation

Λ(s)=(4000s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(4000s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 40004000    =    25⋅532^{5} \cdot 5^{3}
Sign: 11
Analytic conductor: 31.940131.9401
Root analytic conductor: 5.651565.65156
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 4000, ( :1/2), 1)(2,\ 4000,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.1741631891.174163189
L(12)L(\frac12) ≈\approx 1.1741631891.174163189
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 1
good3 1+2.71T+3T2 1 + 2.71T + 3T^{2}
7 1+0.0156T+7T2 1 + 0.0156T + 7T^{2}
11 1−6.41T+11T2 1 - 6.41T + 11T^{2}
13 1+2.76T+13T2 1 + 2.76T + 13T^{2}
17 1+2.07T+17T2 1 + 2.07T + 17T^{2}
19 1−4.25T+19T2 1 - 4.25T + 19T^{2}
23 1−5.97T+23T2 1 - 5.97T + 23T^{2}
29 1−5.37T+29T2 1 - 5.37T + 29T^{2}
31 1+3.59T+31T2 1 + 3.59T + 31T^{2}
37 1−9.04T+37T2 1 - 9.04T + 37T^{2}
41 1+7.03T+41T2 1 + 7.03T + 41T^{2}
43 1+4.96T+43T2 1 + 4.96T + 43T^{2}
47 1−5.73T+47T2 1 - 5.73T + 47T^{2}
53 1+10.5T+53T2 1 + 10.5T + 53T^{2}
59 1−1.13T+59T2 1 - 1.13T + 59T^{2}
61 1−8.76T+61T2 1 - 8.76T + 61T^{2}
67 1−11.0T+67T2 1 - 11.0T + 67T^{2}
71 1+11.0T+71T2 1 + 11.0T + 71T^{2}
73 1+7.38T+73T2 1 + 7.38T + 73T^{2}
79 1−13.4T+79T2 1 - 13.4T + 79T^{2}
83 1−3.44T+83T2 1 - 3.44T + 83T^{2}
89 1+9.49T+89T2 1 + 9.49T + 89T^{2}
97 1−9.21T+97T2 1 - 9.21T + 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

−8.516081620375125130109334771783, −7.40233536277396324035988824057, −6.67999610935587256834527495147, −6.43698634337159075224958741171, −5.42244116753834411195316403958, −4.82942916001445891427704759402, −4.12030512100496072234391150045, −3.05085770675102854732665702675, −1.57103225844582139590880022156, −0.72606013535257107177248168773, 0.72606013535257107177248168773, 1.57103225844582139590880022156, 3.05085770675102854732665702675, 4.12030512100496072234391150045, 4.82942916001445891427704759402, 5.42244116753834411195316403958, 6.43698634337159075224958741171, 6.67999610935587256834527495147, 7.40233536277396324035988824057, 8.516081620375125130109334771783

Graph of the ZZ-function along the critical line