Properties

Label 2-1620-1620.1339-c0-0-1
Degree $2$
Conductor $1620$
Sign $-0.740 + 0.672i$
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  + (−0.286 − 0.957i)2-s + (0.597 − 0.802i)3-s + (−0.835 + 0.549i)4-s + (0.396 + 0.918i)5-s + (−0.939 − 0.342i)6-s + (0.115 − 1.98i)7-s + (0.766 + 0.642i)8-s + (−0.286 − 0.957i)9-s + (0.766 − 0.642i)10-s + (−0.0581 + 0.998i)12-s + (−1.93 + 0.458i)14-s + (0.973 + 0.230i)15-s + (0.396 − 0.918i)16-s + (−0.835 + 0.549i)18-s + (−0.835 − 0.549i)20-s + (−1.52 − 1.27i)21-s + ⋯
L(s)  = 1  + (−0.286 − 0.957i)2-s + (0.597 − 0.802i)3-s + (−0.835 + 0.549i)4-s + (0.396 + 0.918i)5-s + (−0.939 − 0.342i)6-s + (0.115 − 1.98i)7-s + (0.766 + 0.642i)8-s + (−0.286 − 0.957i)9-s + (0.766 − 0.642i)10-s + (−0.0581 + 0.998i)12-s + (−1.93 + 0.458i)14-s + (0.973 + 0.230i)15-s + (0.396 − 0.918i)16-s + (−0.835 + 0.549i)18-s + (−0.835 − 0.549i)20-s + (−1.52 − 1.27i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.740 + 0.672i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.740 + 0.672i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $-0.740 + 0.672i$
Analytic conductor: \(0.808485\)
Root analytic conductor: \(0.899158\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (1339, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :0),\ -0.740 + 0.672i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.138371748\)
\(L(\frac12)\) \(\approx\) \(1.138371748\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.286 + 0.957i)T \)
3 \( 1 + (-0.597 + 0.802i)T \)
5 \( 1 + (-0.396 - 0.918i)T \)
good7 \( 1 + (-0.115 + 1.98i)T + (-0.993 - 0.116i)T^{2} \)
11 \( 1 + (0.286 - 0.957i)T^{2} \)
13 \( 1 + (0.0581 - 0.998i)T^{2} \)
17 \( 1 + (0.939 - 0.342i)T^{2} \)
19 \( 1 + (0.939 + 0.342i)T^{2} \)
23 \( 1 + (-0.00676 - 0.116i)T + (-0.993 + 0.116i)T^{2} \)
29 \( 1 + (-1.73 - 0.412i)T + (0.893 + 0.448i)T^{2} \)
31 \( 1 + (-0.597 - 0.802i)T^{2} \)
37 \( 1 + (-0.766 - 0.642i)T^{2} \)
41 \( 1 + (-0.393 + 1.31i)T + (-0.835 - 0.549i)T^{2} \)
43 \( 1 + (1.52 - 0.177i)T + (0.973 - 0.230i)T^{2} \)
47 \( 1 + (0.512 - 0.257i)T + (0.597 - 0.802i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.286 + 0.957i)T^{2} \)
61 \( 1 + (0.997 + 0.656i)T + (0.396 + 0.918i)T^{2} \)
67 \( 1 + (-1.89 + 0.448i)T + (0.893 - 0.448i)T^{2} \)
71 \( 1 + (-0.173 + 0.984i)T^{2} \)
73 \( 1 + (-0.173 - 0.984i)T^{2} \)
79 \( 1 + (0.835 - 0.549i)T^{2} \)
83 \( 1 + (-0.393 - 1.31i)T + (-0.835 + 0.549i)T^{2} \)
89 \( 1 + (0.439 + 0.368i)T + (0.173 + 0.984i)T^{2} \)
97 \( 1 + (0.686 + 0.727i)T^{2} \)
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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

−9.531473626740412526151150631225, −8.419240879657696071503060967981, −7.73463226871897895727424021203, −7.05909655262213928757743131057, −6.43635800463329259624649523092, −4.85144713881267776670462000797, −3.74272829023577232788115549353, −3.20434694206702390984273578106, −2.05734214625405804406309294466, −0.991462626967129498985460659260, 1.84117353542762479266072955373, 2.99130120805570453415708010534, 4.49376011256734676647488967996, 5.02805541739001838853390845509, 5.73378536077026555573827902331, 6.46439462626130838772597362168, 7.972366883882439690244215199309, 8.493456796973468096205154491952, 8.854748386747692765417821633247, 9.676212252019975825349710551714

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