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Properties

Label 2-1620-180.139-c0-0-2
Degree $2$
Conductor $1620$
Sign $0.642 + 0.766i$
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  − i·2-s − 4-s + (0.866 + 0.5i)5-s + i·8-s + (0.5 − 0.866i)10-s + 16-s − i·17-s + 1.73i·19-s + (−0.866 − 0.5i)20-s + (0.866 − 1.5i)23-s + (0.499 + 0.866i)25-s + (1.5 + 0.866i)31-s − i·32-s − 34-s + 1.73·38-s + ⋯
L(s)  = 1  − i·2-s − 4-s + (0.866 + 0.5i)5-s + i·8-s + (0.5 − 0.866i)10-s + 16-s − i·17-s + 1.73i·19-s + (−0.866 − 0.5i)20-s + (0.866 − 1.5i)23-s + (0.499 + 0.866i)25-s + (1.5 + 0.866i)31-s − i·32-s − 34-s + 1.73·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $0.642 + 0.766i$
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: \((2,\ 1620,\ (\ :0),\ 0.642 + 0.766i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.189646767\)
\(L(\frac12)\) \(\approx\) \(1.189646767\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
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

−9.713772686583426004309441870354, −8.884610130908635367295614016158, −8.156247193233805276416541505646, −7.03178342043208984133608340363, −6.11578502090099886394843767376, −5.24645850017407491451162262342, −4.38597471077108214470574992390, −3.18234470034622892115425425928, −2.49858949313619016988106553230, −1.32091325848621284155959167961, 1.21572576887448141047538343544, 2.78164025556725564333853568695, 4.13788599992158762206052105524, 4.95648371716187288054778374371, 5.68526014918180124435450259358, 6.43831203277864237896773909676, 7.20730112763425922730718025830, 8.141423354622622129630523591242, 8.896884107671611470810025177958, 9.446550360166492978141653216512

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