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Properties

Label 2-1620-1620.1519-c0-0-1
Degree $2$
Conductor $1620$
Sign $0.925 - 0.378i$
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  + (0.993 − 0.116i)2-s + (0.0581 + 0.998i)3-s + (0.973 − 0.230i)4-s + (0.893 + 0.448i)5-s + (0.173 + 0.984i)6-s + (−0.479 − 1.60i)7-s + (0.939 − 0.342i)8-s + (−0.993 + 0.116i)9-s + (0.939 + 0.342i)10-s + (0.286 + 0.957i)12-s + (−0.661 − 1.53i)14-s + (−0.396 + 0.918i)15-s + (0.893 − 0.448i)16-s + (−0.973 + 0.230i)18-s + (0.973 + 0.230i)20-s + (1.57 − 0.571i)21-s + ⋯
L(s)  = 1  + (0.993 − 0.116i)2-s + (0.0581 + 0.998i)3-s + (0.973 − 0.230i)4-s + (0.893 + 0.448i)5-s + (0.173 + 0.984i)6-s + (−0.479 − 1.60i)7-s + (0.939 − 0.342i)8-s + (−0.993 + 0.116i)9-s + (0.939 + 0.342i)10-s + (0.286 + 0.957i)12-s + (−0.661 − 1.53i)14-s + (−0.396 + 0.918i)15-s + (0.893 − 0.448i)16-s + (−0.973 + 0.230i)18-s + (0.973 + 0.230i)20-s + (1.57 − 0.571i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.925 - 0.378i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.925 - 0.378i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $0.925 - 0.378i$
Analytic conductor: \(0.808485\)
Root analytic conductor: \(0.899158\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (1519, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :0),\ 0.925 - 0.378i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.261407202\)
\(L(\frac12)\) \(\approx\) \(2.261407202\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.993 + 0.116i)T \)
3 \( 1 + (-0.0581 - 0.998i)T \)
5 \( 1 + (-0.893 - 0.448i)T \)
good7 \( 1 + (0.479 + 1.60i)T + (-0.835 + 0.549i)T^{2} \)
11 \( 1 + (0.993 + 0.116i)T^{2} \)
13 \( 1 + (0.286 + 0.957i)T^{2} \)
17 \( 1 + (-0.173 + 0.984i)T^{2} \)
19 \( 1 + (-0.173 - 0.984i)T^{2} \)
23 \( 1 + (0.164 - 0.549i)T + (-0.835 - 0.549i)T^{2} \)
29 \( 1 + (0.543 - 1.26i)T + (-0.686 - 0.727i)T^{2} \)
31 \( 1 + (0.0581 - 0.998i)T^{2} \)
37 \( 1 + (0.939 - 0.342i)T^{2} \)
41 \( 1 + (1.18 + 0.138i)T + (0.973 + 0.230i)T^{2} \)
43 \( 1 + (1.57 + 1.03i)T + (0.396 + 0.918i)T^{2} \)
47 \( 1 + (1.36 - 1.44i)T + (-0.0581 - 0.998i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.993 - 0.116i)T^{2} \)
61 \( 1 + (0.113 + 0.0268i)T + (0.893 + 0.448i)T^{2} \)
67 \( 1 + (0.313 + 0.727i)T + (-0.686 + 0.727i)T^{2} \)
71 \( 1 + (-0.766 - 0.642i)T^{2} \)
73 \( 1 + (-0.766 + 0.642i)T^{2} \)
79 \( 1 + (-0.973 + 0.230i)T^{2} \)
83 \( 1 + (-1.18 + 0.138i)T + (0.973 - 0.230i)T^{2} \)
89 \( 1 + (-1.86 + 0.679i)T + (0.766 - 0.642i)T^{2} \)
97 \( 1 + (-0.597 + 0.802i)T^{2} \)
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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

−9.983771529130466224919031434824, −9.181280047110250166969300488777, −7.83729516767770279379988107488, −6.92813141676569223575610820301, −6.33813145557303362595520144456, −5.33344897532830016533668503072, −4.65585874658461799308988191219, −3.54761167404739261839020791080, −3.22238071756158728606803423539, −1.72558122793368821996662736623, 1.80131982772769315839954631768, 2.41761006331175895848521022158, 3.33201438369551394842150375634, 4.93341156872336620560434872368, 5.53456256069217385774179955398, 6.30932083567157386972871416814, 6.63722661402498132259630875565, 7.978652965519999561085681316706, 8.554780907435042550721868138418, 9.431375845411377436603214146181

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