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Properties

Label 2-87e2-1.1-c1-0-242
Degree 22
Conductor 75697569
Sign −1-1
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 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s + 4.63·7-s − 5.69·13-s + 4·16-s + 4.28·19-s − 5·25-s − 9.26·28-s − 8.64·31-s + 11.9·37-s − 13.0·43-s + 14.4·49-s + 11.3·52-s − 4.21·61-s − 8·64-s − 1.77·67-s − 16.2·73-s − 8.56·76-s − 3.39·79-s − 26.3·91-s − 17.5·97-s + 10·100-s + 2.88·103-s + 16.7·109-s + 18.5·112-s + ⋯
L(s)  = 1  − 4-s + 1.75·7-s − 1.57·13-s + 16-s + 0.982·19-s − 25-s − 1.75·28-s − 1.55·31-s + 1.96·37-s − 1.99·43-s + 2.06·49-s + 1.57·52-s − 0.540·61-s − 64-s − 0.216·67-s − 1.90·73-s − 0.982·76-s − 0.381·79-s − 2.76·91-s − 1.77·97-s + 100-s + 0.284·103-s + 1.60·109-s + 1.75·112-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: −1-1
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: 11
Selberg data: (2, 7569, ( :1/2), −1)(2,\ 7569,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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+2T2 1 + 2T^{2}
5 1+5T2 1 + 5T^{2}
7 1−4.63T+7T2 1 - 4.63T + 7T^{2}
11 1+11T2 1 + 11T^{2}
13 1+5.69T+13T2 1 + 5.69T + 13T^{2}
17 1+17T2 1 + 17T^{2}
19 1−4.28T+19T2 1 - 4.28T + 19T^{2}
23 1+23T2 1 + 23T^{2}
31 1+8.64T+31T2 1 + 8.64T + 31T^{2}
37 1−11.9T+37T2 1 - 11.9T + 37T^{2}
41 1+41T2 1 + 41T^{2}
43 1+13.0T+43T2 1 + 13.0T + 43T^{2}
47 1+47T2 1 + 47T^{2}
53 1+53T2 1 + 53T^{2}
59 1+59T2 1 + 59T^{2}
61 1+4.21T+61T2 1 + 4.21T + 61T^{2}
67 1+1.77T+67T2 1 + 1.77T + 67T^{2}
71 1+71T2 1 + 71T^{2}
73 1+16.2T+73T2 1 + 16.2T + 73T^{2}
79 1+3.39T+79T2 1 + 3.39T + 79T^{2}
83 1+83T2 1 + 83T^{2}
89 1+89T2 1 + 89T^{2}
97 1+17.5T+97T2 1 + 17.5T + 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

−7.66346692297959276880566685506, −7.17746938006072002728213330433, −5.80301203246656688854737903269, −5.30432906245248856915194150942, −4.69190740101543008274899999648, −4.22779291434716124932379545367, −3.18458874315738660889833243891, −2.10436212484739088754172804614, −1.28486526853184290613784953983, 0, 1.28486526853184290613784953983, 2.10436212484739088754172804614, 3.18458874315738660889833243891, 4.22779291434716124932379545367, 4.69190740101543008274899999648, 5.30432906245248856915194150942, 5.80301203246656688854737903269, 7.17746938006072002728213330433, 7.66346692297959276880566685506

Graph of the ZZ-function along the critical line