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Properties

Label 2-3920-20.19-c0-0-6
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

\(L(\frac{1}{2})\) \(\approx\) \(2.425203920\)
\(L(\frac12)\) \(\approx\) \(2.425203920\)
\(L(1)\) 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

−8.620934144896486810533805281542, −8.116309383987843399647398539288, −7.45107028408182476548797906625, −6.51703713663024606094925999794, −6.09975052674398827965950187913, −4.66629496976202453060037807522, −3.64394913070110952399276718580, −3.33550101485453386116903008289, −2.48917894556172260893005336881, −1.59646771079412109620937701749, 1.29618632104580819877361630414, 2.24979900847689935477682389354, 2.97064759651776962625400815123, 3.97851471962543880594130770281, 4.68206754014833178583060037693, 5.28886286155025898388959890409, 6.68902735479707347572189483672, 7.41544422845730055219745958040, 7.978018907108165640814119507960, 8.537218119203944927726220286376

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