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Properties

Label 2-1620-180.139-c0-0-4
Degree $2$
Conductor $1620$
Sign $-0.342 + 0.939i$
Analytic cond. $0.808485$
Root an. cond. $0.899158$
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  + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s + (−0.866 − 0.5i)5-s − 0.999i·8-s + (0.499 + 0.866i)10-s + (−0.5 + 0.866i)16-s + i·17-s − 1.73i·19-s − 0.999i·20-s + (0.866 − 1.5i)23-s + (0.499 + 0.866i)25-s + (−1.5 − 0.866i)31-s + (0.866 − 0.499i)32-s + (0.5 − 0.866i)34-s + (−0.866 + 1.49i)38-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s + (−0.866 − 0.5i)5-s − 0.999i·8-s + (0.499 + 0.866i)10-s + (−0.5 + 0.866i)16-s + i·17-s − 1.73i·19-s − 0.999i·20-s + (0.866 − 1.5i)23-s + (0.499 + 0.866i)25-s + (−1.5 − 0.866i)31-s + (0.866 − 0.499i)32-s + (0.5 − 0.866i)34-s + (−0.866 + 1.49i)38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $-0.342 + 0.939i$
Analytic conductor: \(0.808485\)
Root analytic conductor: \(0.899158\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (919, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :0),\ -0.342 + 0.939i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5235269731\)
\(L(\frac12)\) \(\approx\) \(0.5235269731\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 \)
5 \( 1 + (0.866 + 0.5i)T \)
good7 \( 1 + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 - iT - T^{2} \)
19 \( 1 + 1.73iT - T^{2} \)
23 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + iT - T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (0.5 - 0.866i)T^{2} \)
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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

−9.027338435747277268497342666215, −8.809813368347365231525927252282, −7.913195961915888939087012556348, −7.18336866331979119021317794905, −6.42495784055379098722209689995, −5.01144223424448215301516582066, −4.13804765606483582796806651783, −3.22998638764014325221449549178, −2.07937729064730881951366431317, −0.57880136211724557610178476678, 1.40728377320323472824327332004, 2.85903803964139520877266130723, 3.85641075320748535554128539130, 5.14105048468475931556667608079, 5.89453478103287152027248946679, 6.99475320269487844630952026441, 7.44254195846673358088772690772, 8.104468027990694795850056754369, 9.030757442767776555029856085989, 9.666127583289390323113942143497

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