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Properties

Label 2-4896-1.1-c1-0-31
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  + 1.27·5-s − 1.11·7-s + 5.36·11-s + 0.160·13-s + 17-s + 7.36·19-s − 3.92·23-s − 3.38·25-s + 5.06·29-s − 3.06·31-s − 1.41·35-s + 5.06·37-s + 8.01·41-s + 1.18·43-s + 2.54·47-s − 5.76·49-s − 10.1·53-s + 6.81·55-s + 6.17·59-s − 10.7·61-s + 0.204·65-s + 15.8·67-s + 0.845·71-s − 3.95·73-s − 5.95·77-s + 2.56·79-s + 0.222·83-s + ⋯
L(s)  = 1  + 0.568·5-s − 0.419·7-s + 1.61·11-s + 0.0445·13-s + 0.242·17-s + 1.68·19-s − 0.819·23-s − 0.676·25-s + 0.940·29-s − 0.550·31-s − 0.238·35-s + 0.833·37-s + 1.25·41-s + 0.180·43-s + 0.370·47-s − 0.823·49-s − 1.39·53-s + 0.919·55-s + 0.804·59-s − 1.37·61-s + 0.0253·65-s + 1.93·67-s + 0.100·71-s − 0.463·73-s − 0.678·77-s + 0.288·79-s + 0.0243·83-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 2.4772801492.477280149
L(12)L(\frac12) ≈\approx 2.4772801492.477280149
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 1−1.27T+5T2 1 - 1.27T + 5T^{2}
7 1+1.11T+7T2 1 + 1.11T + 7T^{2}
11 1−5.36T+11T2 1 - 5.36T + 11T^{2}
13 1−0.160T+13T2 1 - 0.160T + 13T^{2}
19 1−7.36T+19T2 1 - 7.36T + 19T^{2}
23 1+3.92T+23T2 1 + 3.92T + 23T^{2}
29 1−5.06T+29T2 1 - 5.06T + 29T^{2}
31 1+3.06T+31T2 1 + 3.06T + 31T^{2}
37 1−5.06T+37T2 1 - 5.06T + 37T^{2}
41 1−8.01T+41T2 1 - 8.01T + 41T^{2}
43 1−1.18T+43T2 1 - 1.18T + 43T^{2}
47 1−2.54T+47T2 1 - 2.54T + 47T^{2}
53 1+10.1T+53T2 1 + 10.1T + 53T^{2}
59 1−6.17T+59T2 1 - 6.17T + 59T^{2}
61 1+10.7T+61T2 1 + 10.7T + 61T^{2}
67 1−15.8T+67T2 1 - 15.8T + 67T^{2}
71 1−0.845T+71T2 1 - 0.845T + 71T^{2}
73 1+3.95T+73T2 1 + 3.95T + 73T^{2}
79 1−2.56T+79T2 1 - 2.56T + 79T^{2}
83 1−0.222T+83T2 1 - 0.222T + 83T^{2}
89 1+8.72T+89T2 1 + 8.72T + 89T^{2}
97 1−1.09T+97T2 1 - 1.09T + 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

−8.224498308591763932454799355142, −7.52853103985785809397882221018, −6.70494270583237950823017541355, −6.10386138821250588935781384994, −5.53189169470535493815092562702, −4.47487361253582653359916507610, −3.72204022773422395990315327807, −2.93875327930849934216406510843, −1.80123153949319324885959967794, −0.918510847933424602927354754692, 0.918510847933424602927354754692, 1.80123153949319324885959967794, 2.93875327930849934216406510843, 3.72204022773422395990315327807, 4.47487361253582653359916507610, 5.53189169470535493815092562702, 6.10386138821250588935781384994, 6.70494270583237950823017541355, 7.52853103985785809397882221018, 8.224498308591763932454799355142

Graph of the ZZ-function along the critical line