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Properties

Label 2-87e2-1.1-c1-0-172
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  − 1.15·2-s − 0.667·4-s + 3.44·5-s + 4.51·7-s + 3.07·8-s − 3.98·10-s + 0.554·11-s + 2.75·13-s − 5.21·14-s − 2.21·16-s + 2.99·17-s − 0.278·19-s − 2.30·20-s − 0.639·22-s − 0.927·23-s + 6.89·25-s − 3.18·26-s − 3.01·28-s + 9.12·31-s − 3.59·32-s − 3.45·34-s + 15.5·35-s − 6.87·37-s + 0.321·38-s + 10.6·40-s + 2.85·41-s − 11.2·43-s + ⋯
L(s)  = 1  − 0.816·2-s − 0.333·4-s + 1.54·5-s + 1.70·7-s + 1.08·8-s − 1.25·10-s + 0.167·11-s + 0.765·13-s − 1.39·14-s − 0.554·16-s + 0.726·17-s − 0.0638·19-s − 0.514·20-s − 0.136·22-s − 0.193·23-s + 1.37·25-s − 0.624·26-s − 0.569·28-s + 1.63·31-s − 0.635·32-s − 0.593·34-s + 2.63·35-s − 1.13·37-s + 0.0520·38-s + 1.67·40-s + 0.445·41-s − 1.71·43-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 2.3865148362.386514836
L(12)L(\frac12) ≈\approx 2.3865148362.386514836
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+1.15T+2T2 1 + 1.15T + 2T^{2}
5 1−3.44T+5T2 1 - 3.44T + 5T^{2}
7 1−4.51T+7T2 1 - 4.51T + 7T^{2}
11 1−0.554T+11T2 1 - 0.554T + 11T^{2}
13 1−2.75T+13T2 1 - 2.75T + 13T^{2}
17 1−2.99T+17T2 1 - 2.99T + 17T^{2}
19 1+0.278T+19T2 1 + 0.278T + 19T^{2}
23 1+0.927T+23T2 1 + 0.927T + 23T^{2}
31 1−9.12T+31T2 1 - 9.12T + 31T^{2}
37 1+6.87T+37T2 1 + 6.87T + 37T^{2}
41 1−2.85T+41T2 1 - 2.85T + 41T^{2}
43 1+11.2T+43T2 1 + 11.2T + 43T^{2}
47 1+4.19T+47T2 1 + 4.19T + 47T^{2}
53 1−2.07T+53T2 1 - 2.07T + 53T^{2}
59 1−14.2T+59T2 1 - 14.2T + 59T^{2}
61 1+7.23T+61T2 1 + 7.23T + 61T^{2}
67 1+12.2T+67T2 1 + 12.2T + 67T^{2}
71 1−2.38T+71T2 1 - 2.38T + 71T^{2}
73 1+0.913T+73T2 1 + 0.913T + 73T^{2}
79 1−2.06T+79T2 1 - 2.06T + 79T^{2}
83 1−10.9T+83T2 1 - 10.9T + 83T^{2}
89 1−6.80T+89T2 1 - 6.80T + 89T^{2}
97 1+3.51T+97T2 1 + 3.51T + 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

−8.156776127075500786389965093130, −7.40066314358144070447698494444, −6.49077659362897102245790593901, −5.75036739227558090743025034871, −5.03670189713552388540652349546, −4.62112533165619204994759912178, −3.51963730269687316593241819313, −2.21897371772611866937310066631, −1.56890303528569594701398545878, −1.01543328522474359073883645238, 1.01543328522474359073883645238, 1.56890303528569594701398545878, 2.21897371772611866937310066631, 3.51963730269687316593241819313, 4.62112533165619204994759912178, 5.03670189713552388540652349546, 5.75036739227558090743025034871, 6.49077659362897102245790593901, 7.40066314358144070447698494444, 8.156776127075500786389965093130

Graph of the ZZ-function along the critical line