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Properties

Label 2-1620-1620.139-c0-0-0
Degree $2$
Conductor $1620$
Sign $-0.790 + 0.612i$
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  + (−0.597 + 0.802i)2-s + (−0.893 + 0.448i)3-s + (−0.286 − 0.957i)4-s + (−0.835 − 0.549i)5-s + (0.173 − 0.984i)6-s + (−0.0798 − 0.0845i)7-s + (0.939 + 0.342i)8-s + (0.597 − 0.802i)9-s + (0.939 − 0.342i)10-s + (0.686 + 0.727i)12-s + (0.115 − 0.0135i)14-s + (0.993 + 0.116i)15-s + (−0.835 + 0.549i)16-s + (0.286 + 0.957i)18-s + (−0.286 + 0.957i)20-s + (0.109 + 0.0397i)21-s + ⋯
L(s)  = 1  + (−0.597 + 0.802i)2-s + (−0.893 + 0.448i)3-s + (−0.286 − 0.957i)4-s + (−0.835 − 0.549i)5-s + (0.173 − 0.984i)6-s + (−0.0798 − 0.0845i)7-s + (0.939 + 0.342i)8-s + (0.597 − 0.802i)9-s + (0.939 − 0.342i)10-s + (0.686 + 0.727i)12-s + (0.115 − 0.0135i)14-s + (0.993 + 0.116i)15-s + (−0.835 + 0.549i)16-s + (0.286 + 0.957i)18-s + (−0.286 + 0.957i)20-s + (0.109 + 0.0397i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.790 + 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.790 + 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $-0.790 + 0.612i$
Analytic conductor: \(0.808485\)
Root analytic conductor: \(0.899158\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :0),\ -0.790 + 0.612i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02645540635\)
\(L(\frac12)\) \(\approx\) \(0.02645540635\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.597 - 0.802i)T \)
3 \( 1 + (0.893 - 0.448i)T \)
5 \( 1 + (0.835 + 0.549i)T \)
good7 \( 1 + (0.0798 + 0.0845i)T + (-0.0581 + 0.998i)T^{2} \)
11 \( 1 + (-0.597 - 0.802i)T^{2} \)
13 \( 1 + (0.686 + 0.727i)T^{2} \)
17 \( 1 + (-0.173 - 0.984i)T^{2} \)
19 \( 1 + (-0.173 + 0.984i)T^{2} \)
23 \( 1 + (0.941 - 0.998i)T + (-0.0581 - 0.998i)T^{2} \)
29 \( 1 + (1.93 + 0.225i)T + (0.973 + 0.230i)T^{2} \)
31 \( 1 + (-0.893 - 0.448i)T^{2} \)
37 \( 1 + (0.939 + 0.342i)T^{2} \)
41 \( 1 + (-0.473 - 0.635i)T + (-0.286 + 0.957i)T^{2} \)
43 \( 1 + (0.109 + 1.87i)T + (-0.993 + 0.116i)T^{2} \)
47 \( 1 + (1.16 - 0.275i)T + (0.893 - 0.448i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (-0.597 + 0.802i)T^{2} \)
61 \( 1 + (0.512 - 1.71i)T + (-0.835 - 0.549i)T^{2} \)
67 \( 1 + (1.97 - 0.230i)T + (0.973 - 0.230i)T^{2} \)
71 \( 1 + (-0.766 + 0.642i)T^{2} \)
73 \( 1 + (-0.766 - 0.642i)T^{2} \)
79 \( 1 + (0.286 + 0.957i)T^{2} \)
83 \( 1 + (0.473 - 0.635i)T + (-0.286 - 0.957i)T^{2} \)
89 \( 1 + (1.12 + 0.408i)T + (0.766 + 0.642i)T^{2} \)
97 \( 1 + (-0.396 + 0.918i)T^{2} \)
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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

−9.883813652466902675005783971159, −9.287032066733037076052297852432, −8.478991662305057897947441836181, −7.55087899690956133244092063644, −7.04247656676880857905511792982, −5.88442677669936773828479401551, −5.40542855318649953963671623994, −4.41158632130823975307912315639, −3.71210430059757636389140848850, −1.51347069553675342059029251095, 0.02941273261799445441784069481, 1.68364094756523776466465752080, 2.84912343429845540594534758869, 3.95451332792459124249801120730, 4.72706938072974732477275654894, 6.01865968907002303614924713177, 6.86076779526490946210426278002, 7.67977726663118723904634289788, 8.124553727789007233244879254921, 9.260107774613005950746261895259

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