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Properties

Label 2-4896-1.1-c1-0-38
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  + 1.22·5-s + 4.88·7-s + 5.50·11-s − 4.95·13-s + 17-s − 1.70·19-s + 0.614·23-s − 3.50·25-s + 8.17·29-s + 4.88·31-s + 5.98·35-s + 7.27·37-s − 6.50·41-s + 7.68·43-s − 13.1·47-s + 16.9·49-s − 9.45·53-s + 6.73·55-s + 2.56·59-s + 9.72·61-s − 6.05·65-s + 5.98·67-s − 2.70·71-s + 6·73-s + 26.9·77-s − 6.11·79-s − 2.56·83-s + ⋯
L(s)  = 1  + 0.547·5-s + 1.84·7-s + 1.65·11-s − 1.37·13-s + 0.242·17-s − 0.391·19-s + 0.128·23-s − 0.700·25-s + 1.51·29-s + 0.878·31-s + 1.01·35-s + 1.19·37-s − 1.01·41-s + 1.17·43-s − 1.92·47-s + 2.41·49-s − 1.29·53-s + 0.907·55-s + 0.334·59-s + 1.24·61-s − 0.751·65-s + 0.730·67-s − 0.320·71-s + 0.702·73-s + 3.06·77-s − 0.688·79-s − 0.281·83-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 3.0820696953.082069695
L(12)L(\frac12) ≈\approx 3.0820696953.082069695
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 1−1.22T+5T2 1 - 1.22T + 5T^{2}
7 1−4.88T+7T2 1 - 4.88T + 7T^{2}
11 1−5.50T+11T2 1 - 5.50T + 11T^{2}
13 1+4.95T+13T2 1 + 4.95T + 13T^{2}
19 1+1.70T+19T2 1 + 1.70T + 19T^{2}
23 1−0.614T+23T2 1 - 0.614T + 23T^{2}
29 1−8.17T+29T2 1 - 8.17T + 29T^{2}
31 1−4.88T+31T2 1 - 4.88T + 31T^{2}
37 1−7.27T+37T2 1 - 7.27T + 37T^{2}
41 1+6.50T+41T2 1 + 6.50T + 41T^{2}
43 1−7.68T+43T2 1 - 7.68T + 43T^{2}
47 1+13.1T+47T2 1 + 13.1T + 47T^{2}
53 1+9.45T+53T2 1 + 9.45T + 53T^{2}
59 1−2.56T+59T2 1 - 2.56T + 59T^{2}
61 1−9.72T+61T2 1 - 9.72T + 61T^{2}
67 1−5.98T+67T2 1 - 5.98T + 67T^{2}
71 1+2.70T+71T2 1 + 2.70T + 71T^{2}
73 1−6T+73T2 1 - 6T + 73T^{2}
79 1+6.11T+79T2 1 + 6.11T + 79T^{2}
83 1+2.56T+83T2 1 + 2.56T + 83T^{2}
89 1−7.90T+89T2 1 - 7.90T + 89T^{2}
97 1+5.45T+97T2 1 + 5.45T + 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

−8.212781197347460990873643747028, −7.68248203594612559906523254586, −6.75195677570493690292492333769, −6.18071988717769782685552891187, −5.12064643304895436118150250164, −4.68399968409844911430856790333, −3.96703988555256437580870744282, −2.63805293893139868379622401029, −1.82825355735503630508362639183, −1.06264943983690810650628030253, 1.06264943983690810650628030253, 1.82825355735503630508362639183, 2.63805293893139868379622401029, 3.96703988555256437580870744282, 4.68399968409844911430856790333, 5.12064643304895436118150250164, 6.18071988717769782685552891187, 6.75195677570493690292492333769, 7.68248203594612559906523254586, 8.212781197347460990873643747028

Graph of the ZZ-function along the critical line