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Properties

Label 2-1620-180.79-c0-0-2
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 − 0.999·10-s + (−0.5 + 0.866i)16-s + 2i·17-s + (−0.866 − 0.499i)20-s + (0.499 − 0.866i)25-s + (−0.866 + 0.499i)32-s + (−1 + 1.73i)34-s + (−0.499 − 0.866i)40-s + (0.5 + 0.866i)49-s + (0.866 − 0.499i)50-s − 2i·53-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 − 0.999·10-s + (−0.5 + 0.866i)16-s + 2i·17-s + (−0.866 − 0.499i)20-s + (0.499 − 0.866i)25-s + (−0.866 + 0.499i)32-s + (−1 + 1.73i)34-s + (−0.499 − 0.866i)40-s + (0.5 + 0.866i)49-s + (0.866 − 0.499i)50-s − 2i·53-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: $\chi_{1620} (379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :0),\ -0.342 - 0.939i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.512271517\)
\(L(\frac12)\) \(\approx\) \(1.512271517\)
\(L(1)\) 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 \( 1 - 2iT - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
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 \( 1 + 2iT - T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.5 - 0.866i)T^{2} \)
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

−9.956224672842400579962658046164, −8.572085311240701868569664080843, −8.164791442267288846974001991058, −7.32773550331082389186611026830, −6.54963811539873896149969850818, −5.87940670171164686388235316514, −4.78424812946469498178519331162, −3.91024108092007094341439498897, −3.32885454178301273009373594473, −2.06023936028566122535163056463, 0.925882931185934536573689556123, 2.48602737220232797426061366771, 3.39895092753292895614691590449, 4.36669788582558019045778846385, 4.99548007048139648599869820669, 5.83014395926782614067774759010, 7.04057901194451708312205217036, 7.47412640919528934777026826288, 8.694095520927826334682588040008, 9.403097270269888767349868584639

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