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Properties

Label 2-5376-1.1-c1-0-32
Degree 22
Conductor 53765376
Sign 11
Analytic cond. 42.927542.9275
Root an. cond. 6.551916.55191
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  + 3-s + 0.732·5-s − 7-s + 9-s + 4.19·11-s + 0.732·15-s − 1.26·17-s − 7.46·19-s − 21-s + 8.73·23-s − 4.46·25-s + 27-s + 3.46·29-s − 4.92·31-s + 4.19·33-s − 0.732·35-s + 10·37-s − 1.26·41-s + 0.928·43-s + 0.732·45-s + 6.92·47-s + 49-s − 1.26·51-s + 3.46·53-s + 3.07·55-s − 7.46·57-s + 10.9·59-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.327·5-s − 0.377·7-s + 0.333·9-s + 1.26·11-s + 0.189·15-s − 0.307·17-s − 1.71·19-s − 0.218·21-s + 1.82·23-s − 0.892·25-s + 0.192·27-s + 0.643·29-s − 0.885·31-s + 0.730·33-s − 0.123·35-s + 1.64·37-s − 0.198·41-s + 0.141·43-s + 0.109·45-s + 1.01·47-s + 0.142·49-s − 0.177·51-s + 0.475·53-s + 0.414·55-s − 0.988·57-s + 1.42·59-s + ⋯

Functional equation

Λ(s)=(5376s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 5376 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(5376s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 5376 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 53765376    =    28⋅3⋅72^{8} \cdot 3 \cdot 7
Sign: 11
Analytic conductor: 42.927542.9275
Root analytic conductor: 6.551916.55191
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 5376, ( :1/2), 1)(2,\ 5376,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.7215489062.721548906
L(12)L(\frac12) ≈\approx 2.7215489062.721548906
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−T 1 - T
7 1+T 1 + T
good5 1−0.732T+5T2 1 - 0.732T + 5T^{2}
11 1−4.19T+11T2 1 - 4.19T + 11T^{2}
13 1+13T2 1 + 13T^{2}
17 1+1.26T+17T2 1 + 1.26T + 17T^{2}
19 1+7.46T+19T2 1 + 7.46T + 19T^{2}
23 1−8.73T+23T2 1 - 8.73T + 23T^{2}
29 1−3.46T+29T2 1 - 3.46T + 29T^{2}
31 1+4.92T+31T2 1 + 4.92T + 31T^{2}
37 1−10T+37T2 1 - 10T + 37T^{2}
41 1+1.26T+41T2 1 + 1.26T + 41T^{2}
43 1−0.928T+43T2 1 - 0.928T + 43T^{2}
47 1−6.92T+47T2 1 - 6.92T + 47T^{2}
53 1−3.46T+53T2 1 - 3.46T + 53T^{2}
59 1−10.9T+59T2 1 - 10.9T + 59T^{2}
61 1−1.46T+61T2 1 - 1.46T + 61T^{2}
67 1−7.46T+67T2 1 - 7.46T + 67T^{2}
71 1−14.1T+71T2 1 - 14.1T + 71T^{2}
73 1+10.3T+73T2 1 + 10.3T + 73T^{2}
79 1−9.46T+79T2 1 - 9.46T + 79T^{2}
83 1+8.39T+83T2 1 + 8.39T + 83T^{2}
89 1−1.26T+89T2 1 - 1.26T + 89T^{2}
97 1−6.39T+97T2 1 - 6.39T + 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

−8.376240015135890807307314059851, −7.38561352211010917423906586039, −6.69213780448247960070151225424, −6.25179625482481576241879389631, −5.28200897054093502751754310643, −4.24488603290357350864706653370, −3.82152617405076016790866812527, −2.73613962062443313741311136127, −2.00821155542493407239049129768, −0.882213172867133979406920798167, 0.882213172867133979406920798167, 2.00821155542493407239049129768, 2.73613962062443313741311136127, 3.82152617405076016790866812527, 4.24488603290357350864706653370, 5.28200897054093502751754310643, 6.25179625482481576241879389631, 6.69213780448247960070151225424, 7.38561352211010917423906586039, 8.376240015135890807307314059851

Graph of the ZZ-function along the critical line