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Properties

Label 2-3920-20.19-c0-0-1
Degree $2$
Conductor $3920$
Sign $-0.866 - 0.5i$
Analytic cond. $1.95633$
Root an. cond. $1.39869$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

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

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

Invariants

Degree: \(2\)
Conductor: \(3920\)    =    \(2^{4} \cdot 5 \cdot 7^{2}\)
Sign: $-0.866 - 0.5i$
Analytic conductor: \(1.95633\)
Root analytic conductor: \(1.39869\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3920} (3039, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3920,\ (\ :0),\ -0.866 - 0.5i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5850731938\)
\(L(\frac12)\) \(\approx\) \(0.5850731938\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 - iT \)
7 \( 1 \)
good3 \( 1 + 1.73T + T^{2} \)
11 \( 1 - 1.73iT - T^{2} \)
13 \( 1 - iT - T^{2} \)
17 \( 1 - iT - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 - T + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 - 1.73T + T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + 2iT - T^{2} \)
79 \( 1 - 1.73iT - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 - iT - T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

Graph of the $Z$-function along the critical line