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Properties

Label 2-87e2-1.1-c1-0-142
Degree 22
Conductor 75697569
Sign 11
Analytic cond. 60.438760.4387
Root an. cond. 7.774237.77423
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  − 2.30·2-s + 3.33·4-s + 3.03·5-s − 4.39·7-s − 3.07·8-s − 7.01·10-s + 5.68·11-s + 3.95·13-s + 10.1·14-s + 0.441·16-s + 3.21·17-s + 3.61·19-s + 10.1·20-s − 13.1·22-s + 2.69·23-s + 4.21·25-s − 9.12·26-s − 14.6·28-s + 0.823·31-s + 5.13·32-s − 7.42·34-s − 13.3·35-s + 5.60·37-s − 8.35·38-s − 9.34·40-s − 0.558·41-s + 12.7·43-s + ⋯
L(s)  = 1  − 1.63·2-s + 1.66·4-s + 1.35·5-s − 1.66·7-s − 1.08·8-s − 2.21·10-s + 1.71·11-s + 1.09·13-s + 2.71·14-s + 0.110·16-s + 0.780·17-s + 0.830·19-s + 2.26·20-s − 2.79·22-s + 0.562·23-s + 0.843·25-s − 1.78·26-s − 2.76·28-s + 0.147·31-s + 0.907·32-s − 1.27·34-s − 2.25·35-s + 0.921·37-s − 1.35·38-s − 1.47·40-s − 0.0872·41-s + 1.93·43-s + ⋯

Functional equation

Λ(s)=(7569s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 7569 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(7569s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 7569 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 75697569    =    32⋅2923^{2} \cdot 29^{2}
Sign: 11
Analytic conductor: 60.438760.4387
Root analytic conductor: 7.774237.77423
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 7569, ( :1/2), 1)(2,\ 7569,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.3872495701.387249570
L(12)L(\frac12) ≈\approx 1.3872495701.387249570
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)
bad3 1 1
29 1 1
good2 1+2.30T+2T2 1 + 2.30T + 2T^{2}
5 1−3.03T+5T2 1 - 3.03T + 5T^{2}
7 1+4.39T+7T2 1 + 4.39T + 7T^{2}
11 1−5.68T+11T2 1 - 5.68T + 11T^{2}
13 1−3.95T+13T2 1 - 3.95T + 13T^{2}
17 1−3.21T+17T2 1 - 3.21T + 17T^{2}
19 1−3.61T+19T2 1 - 3.61T + 19T^{2}
23 1−2.69T+23T2 1 - 2.69T + 23T^{2}
31 1−0.823T+31T2 1 - 0.823T + 31T^{2}
37 1−5.60T+37T2 1 - 5.60T + 37T^{2}
41 1+0.558T+41T2 1 + 0.558T + 41T^{2}
43 1−12.7T+43T2 1 - 12.7T + 43T^{2}
47 1+0.129T+47T2 1 + 0.129T + 47T^{2}
53 1+6.88T+53T2 1 + 6.88T + 53T^{2}
59 1−0.745T+59T2 1 - 0.745T + 59T^{2}
61 1−7.17T+61T2 1 - 7.17T + 61T^{2}
67 1−7.06T+67T2 1 - 7.06T + 67T^{2}
71 1−12.6T+71T2 1 - 12.6T + 71T^{2}
73 1+8.81T+73T2 1 + 8.81T + 73T^{2}
79 1−6.99T+79T2 1 - 6.99T + 79T^{2}
83 1+8.90T+83T2 1 + 8.90T + 83T^{2}
89 1+1.51T+89T2 1 + 1.51T + 89T^{2}
97 1+14.7T+97T2 1 + 14.7T + 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

−8.048553425958415110475721728287, −7.10053333789112670761537730720, −6.58505865383784042485928501711, −6.14626728933147170142281301332, −5.55999922590262892004939872348, −4.06084747751950742430034703873, −3.25587017742218836587592774088, −2.43831319009164804275880040193, −1.32718891362557278936359078297, −0.891200533339698870697225726519, 0.891200533339698870697225726519, 1.32718891362557278936359078297, 2.43831319009164804275880040193, 3.25587017742218836587592774088, 4.06084747751950742430034703873, 5.55999922590262892004939872348, 6.14626728933147170142281301332, 6.58505865383784042485928501711, 7.10053333789112670761537730720, 8.048553425958415110475721728287

Graph of the ZZ-function along the critical line