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Properties

Label 2-4896-1.1-c1-0-21
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  + 1.90·5-s − 2.28·7-s + 2.68·11-s − 4.18·13-s − 17-s − 0.689·19-s + 3.39·23-s − 1.38·25-s + 5.86·29-s + 7.86·31-s − 4.34·35-s − 5.86·37-s − 2.61·41-s + 2.88·43-s + 3.80·47-s − 1.77·49-s + 0.423·53-s + 5.11·55-s + 3.57·59-s + 3.63·61-s − 7.96·65-s + 2.64·67-s + 12.4·71-s + 8.14·73-s − 6.14·77-s + 10.0·79-s − 2.57·83-s + ⋯
L(s)  = 1  + 0.850·5-s − 0.864·7-s + 0.810·11-s − 1.16·13-s − 0.242·17-s − 0.158·19-s + 0.708·23-s − 0.277·25-s + 1.08·29-s + 1.41·31-s − 0.734·35-s − 0.963·37-s − 0.407·41-s + 0.440·43-s + 0.554·47-s − 0.253·49-s + 0.0581·53-s + 0.689·55-s + 0.465·59-s + 0.465·61-s − 0.987·65-s + 0.323·67-s + 1.47·71-s + 0.953·73-s − 0.700·77-s + 1.13·79-s − 0.282·83-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 1.9770247961.977024796
L(12)L(\frac12) ≈\approx 1.9770247961.977024796
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 1−1.90T+5T2 1 - 1.90T + 5T^{2}
7 1+2.28T+7T2 1 + 2.28T + 7T^{2}
11 1−2.68T+11T2 1 - 2.68T + 11T^{2}
13 1+4.18T+13T2 1 + 4.18T + 13T^{2}
19 1+0.689T+19T2 1 + 0.689T + 19T^{2}
23 1−3.39T+23T2 1 - 3.39T + 23T^{2}
29 1−5.86T+29T2 1 - 5.86T + 29T^{2}
31 1−7.86T+31T2 1 - 7.86T + 31T^{2}
37 1+5.86T+37T2 1 + 5.86T + 37T^{2}
41 1+2.61T+41T2 1 + 2.61T + 41T^{2}
43 1−2.88T+43T2 1 - 2.88T + 43T^{2}
47 1−3.80T+47T2 1 - 3.80T + 47T^{2}
53 1−0.423T+53T2 1 - 0.423T + 53T^{2}
59 1−3.57T+59T2 1 - 3.57T + 59T^{2}
61 1−3.63T+61T2 1 - 3.63T + 61T^{2}
67 1−2.64T+67T2 1 - 2.64T + 67T^{2}
71 1−12.4T+71T2 1 - 12.4T + 71T^{2}
73 1−8.14T+73T2 1 - 8.14T + 73T^{2}
79 1−10.0T+79T2 1 - 10.0T + 79T^{2}
83 1+2.57T+83T2 1 + 2.57T + 83T^{2}
89 1+7.37T+89T2 1 + 7.37T + 89T^{2}
97 1−4.02T+97T2 1 - 4.02T + 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

−8.401830996302521599060292807745, −7.38961492541064744746868010421, −6.60306845861505100876930847200, −6.34434190361102115135175896451, −5.32274135209037474489527326729, −4.66622988943898236861887273048, −3.67527800322157614430546220648, −2.77215785680720940908655419730, −2.03330794088023983853831735391, −0.76414961271303616860381933082, 0.76414961271303616860381933082, 2.03330794088023983853831735391, 2.77215785680720940908655419730, 3.67527800322157614430546220648, 4.66622988943898236861887273048, 5.32274135209037474489527326729, 6.34434190361102115135175896451, 6.60306845861505100876930847200, 7.38961492541064744746868010421, 8.401830996302521599060292807745

Graph of the ZZ-function along the critical line