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Properties

Label 2-4896-1.1-c1-0-32
Degree 22
Conductor 48964896
Sign 11
Analytic cond. 39.094739.0947
Root an. cond. 6.252586.25258
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  + 2.37·5-s + 2·7-s − 2.37·11-s − 0.372·13-s + 17-s + 6.37·19-s + 4.37·23-s + 0.627·25-s − 4.74·29-s − 2·31-s + 4.74·35-s + 4·37-s − 3.62·41-s + 11.1·43-s + 4·47-s − 3·49-s − 6.74·53-s − 5.62·55-s + 0.744·59-s + 8.74·61-s − 0.883·65-s + 4·67-s + 1.25·71-s + 14.7·73-s − 4.74·77-s − 2·79-s + 12·83-s + ⋯
L(s)  = 1  + 1.06·5-s + 0.755·7-s − 0.715·11-s − 0.103·13-s + 0.242·17-s + 1.46·19-s + 0.911·23-s + 0.125·25-s − 0.881·29-s − 0.359·31-s + 0.801·35-s + 0.657·37-s − 0.566·41-s + 1.69·43-s + 0.583·47-s − 0.428·49-s − 0.926·53-s − 0.758·55-s + 0.0969·59-s + 1.11·61-s − 0.109·65-s + 0.488·67-s + 0.148·71-s + 1.72·73-s − 0.540·77-s − 0.225·79-s + 1.31·83-s + ⋯

Functional equation

Λ(s)=(4896s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(4896s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 48964896    =    25⋅32⋅172^{5} \cdot 3^{2} \cdot 17
Sign: 11
Analytic conductor: 39.094739.0947
Root analytic conductor: 6.252586.25258
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 4896, ( :1/2), 1)(2,\ 4896,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.7543871992.754387199
L(12)L(\frac12) ≈\approx 2.7543871992.754387199
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
17 1−T 1 - T
good5 1−2.37T+5T2 1 - 2.37T + 5T^{2}
7 1−2T+7T2 1 - 2T + 7T^{2}
11 1+2.37T+11T2 1 + 2.37T + 11T^{2}
13 1+0.372T+13T2 1 + 0.372T + 13T^{2}
19 1−6.37T+19T2 1 - 6.37T + 19T^{2}
23 1−4.37T+23T2 1 - 4.37T + 23T^{2}
29 1+4.74T+29T2 1 + 4.74T + 29T^{2}
31 1+2T+31T2 1 + 2T + 31T^{2}
37 1−4T+37T2 1 - 4T + 37T^{2}
41 1+3.62T+41T2 1 + 3.62T + 41T^{2}
43 1−11.1T+43T2 1 - 11.1T + 43T^{2}
47 1−4T+47T2 1 - 4T + 47T^{2}
53 1+6.74T+53T2 1 + 6.74T + 53T^{2}
59 1−0.744T+59T2 1 - 0.744T + 59T^{2}
61 1−8.74T+61T2 1 - 8.74T + 61T^{2}
67 1−4T+67T2 1 - 4T + 67T^{2}
71 1−1.25T+71T2 1 - 1.25T + 71T^{2}
73 1−14.7T+73T2 1 - 14.7T + 73T^{2}
79 1+2T+79T2 1 + 2T + 79T^{2}
83 1−12T+83T2 1 - 12T + 83T^{2}
89 1+15.4T+89T2 1 + 15.4T + 89T^{2}
97 1−15.4T+97T2 1 - 15.4T + 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

−8.117592689300812357640999148276, −7.62055256052759117710335953694, −6.88560652284055589642916146169, −5.86984933770762510253291470084, −5.36648764525211078665313450497, −4.83882275063491798827251642425, −3.67927091955995892445940012300, −2.71610556825917031707046247163, −1.93266109125584714577750789328, −0.954320009094335107604431114378, 0.954320009094335107604431114378, 1.93266109125584714577750789328, 2.71610556825917031707046247163, 3.67927091955995892445940012300, 4.83882275063491798827251642425, 5.36648764525211078665313450497, 5.86984933770762510253291470084, 6.88560652284055589642916146169, 7.62055256052759117710335953694, 8.117592689300812357640999148276

Graph of the ZZ-function along the critical line