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Properties

Label 2-3920-20.19-c0-0-7
Degree 22
Conductor 39203920
Sign −0.866+0.5i-0.866 + 0.5i
Analytic cond. 1.956331.95633
Root an. cond. 1.398691.39869
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.73·3-s − i·5-s + 1.99·9-s − 1.73i·11-s − i·13-s + 1.73i·15-s − i·17-s − 25-s − 1.73·27-s + 29-s + 2.99i·33-s + 1.73i·39-s − 1.99i·45-s + 1.73·47-s + 1.73i·51-s + ⋯
L(s)  = 1  − 1.73·3-s − i·5-s + 1.99·9-s − 1.73i·11-s − i·13-s + 1.73i·15-s − i·17-s − 25-s − 1.73·27-s + 29-s + 2.99i·33-s + 1.73i·39-s − 1.99i·45-s + 1.73·47-s + 1.73i·51-s + ⋯

Functional equation

Λ(s)=(3920s/2 ΓC(s) L(s)=((−0.866+0.5i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 + 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3920s/2 ΓC(s) L(s)=((−0.866+0.5i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 + 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 39203920    =    24⋅5⋅722^{4} \cdot 5 \cdot 7^{2}
Sign: −0.866+0.5i-0.866 + 0.5i
Analytic conductor: 1.956331.95633
Root analytic conductor: 1.398691.39869
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3920(3039,⋅)\chi_{3920} (3039, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3920, ( :0), −0.866+0.5i)(2,\ 3920,\ (\ :0),\ -0.866 + 0.5i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.58507319380.5850731938
L(12)L(\frac12) ≈\approx 0.58507319380.5850731938
L(1)L(1) 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
5 1+iT 1 + iT
7 1 1
good3 1+1.73T+T2 1 + 1.73T + T^{2}
11 1+1.73iT−T2 1 + 1.73iT - T^{2}
13 1+iT−T2 1 + iT - T^{2}
17 1+iT−T2 1 + iT - T^{2}
19 1−T2 1 - T^{2}
23 1+T2 1 + T^{2}
29 1−T+T2 1 - T + T^{2}
31 1−T2 1 - T^{2}
37 1−T2 1 - T^{2}
41 1+T2 1 + T^{2}
43 1+T2 1 + T^{2}
47 1−1.73T+T2 1 - 1.73T + T^{2}
53 1−T2 1 - T^{2}
59 1−T2 1 - T^{2}
61 1+T2 1 + T^{2}
67 1+T2 1 + T^{2}
71 1−T2 1 - T^{2}
73 1−2iT−T2 1 - 2iT - T^{2}
79 1+1.73iT−T2 1 + 1.73iT - T^{2}
83 1+T2 1 + T^{2}
89 1+T2 1 + T^{2}
97 1+iT−T2 1 + iT - T^{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

−8.355439019255283775200337843968, −7.56253527107589767684483342693, −6.64286674807705175566655959991, −5.81441181110851486691691011909, −5.51702390320995492468018590750, −4.84211575406691721862244066989, −3.98375307145101670305929932306, −2.84011768945163756429693933205, −1.12405357878212052987835012073, −0.50673974312971847292206995767, 1.49788453221545786838645193554, 2.39958354495418204134704667338, 3.94389515701335131694057178352, 4.44956608790212105882203027814, 5.29037926303853323696174108194, 6.15302193392129514545646118067, 6.69862715546768388165979670082, 7.12066601210845365189225237981, 7.913735587951438152360833564461, 9.212531110865990179079740084984

Graph of the ZZ-function along the critical line