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Properties

Label 2-72e2-1.1-c1-0-39
Degree 22
Conductor 51845184
Sign 11
Analytic cond. 41.394441.3944
Root an. cond. 6.433856.43385
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  + 2.46·5-s + 6.46·13-s + 5.92·17-s + 1.07·25-s − 1.53·29-s + 9.39·37-s + 10·41-s − 7·49-s − 14·53-s − 15.3·61-s + 15.9·65-s + 16.8·73-s + 14.6·85-s − 18.8·89-s + 18·97-s + 2·101-s − 14.3·109-s + 20.8·113-s + ⋯
L(s)  = 1  + 1.10·5-s + 1.79·13-s + 1.43·17-s + 0.214·25-s − 0.285·29-s + 1.54·37-s + 1.56·41-s − 49-s − 1.92·53-s − 1.97·61-s + 1.97·65-s + 1.97·73-s + 1.58·85-s − 1.99·89-s + 1.82·97-s + 0.199·101-s − 1.37·109-s + 1.96·113-s + ⋯

Functional equation

Λ(s)=(5184s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(5184s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 51845184    =    26⋅342^{6} \cdot 3^{4}
Sign: 11
Analytic conductor: 41.394441.3944
Root analytic conductor: 6.433856.43385
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 5184, ( :1/2), 1)(2,\ 5184,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.9739545592.973954559
L(12)L(\frac12) ≈\approx 2.9739545592.973954559
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
3 1 1
good5 1−2.46T+5T2 1 - 2.46T + 5T^{2}
7 1+7T2 1 + 7T^{2}
11 1+11T2 1 + 11T^{2}
13 1−6.46T+13T2 1 - 6.46T + 13T^{2}
17 1−5.92T+17T2 1 - 5.92T + 17T^{2}
19 1+19T2 1 + 19T^{2}
23 1+23T2 1 + 23T^{2}
29 1+1.53T+29T2 1 + 1.53T + 29T^{2}
31 1+31T2 1 + 31T^{2}
37 1−9.39T+37T2 1 - 9.39T + 37T^{2}
41 1−10T+41T2 1 - 10T + 41T^{2}
43 1+43T2 1 + 43T^{2}
47 1+47T2 1 + 47T^{2}
53 1+14T+53T2 1 + 14T + 53T^{2}
59 1+59T2 1 + 59T^{2}
61 1+15.3T+61T2 1 + 15.3T + 61T^{2}
67 1+67T2 1 + 67T^{2}
71 1+71T2 1 + 71T^{2}
73 1−16.8T+73T2 1 - 16.8T + 73T^{2}
79 1+79T2 1 + 79T^{2}
83 1+83T2 1 + 83T^{2}
89 1+18.8T+89T2 1 + 18.8T + 89T^{2}
97 1−18T+97T2 1 - 18T + 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

−8.077050173209106070326267514398, −7.70624430007089046212799027795, −6.47436308696754808961874685495, −6.04161228800115391918699617432, −5.55759572377750697368283758701, −4.55532150381668773045619297085, −3.61740212162844662839798720138, −2.87993906012818302723350816276, −1.74120521064781890205680362285, −1.03071923477985079780402782356, 1.03071923477985079780402782356, 1.74120521064781890205680362285, 2.87993906012818302723350816276, 3.61740212162844662839798720138, 4.55532150381668773045619297085, 5.55759572377750697368283758701, 6.04161228800115391918699617432, 6.47436308696754808961874685495, 7.70624430007089046212799027795, 8.077050173209106070326267514398

Graph of the ZZ-function along the critical line