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Properties

Label 2-1620-1620.439-c0-0-0
Degree $2$
Conductor $1620$
Sign $-0.135 + 0.990i$
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  + (0.396 − 0.918i)2-s + (−0.835 + 0.549i)3-s + (−0.686 − 0.727i)4-s + (−0.0581 − 0.998i)5-s + (0.173 + 0.984i)6-s + (1.73 + 0.412i)7-s + (−0.939 + 0.342i)8-s + (0.396 − 0.918i)9-s + (−0.939 − 0.342i)10-s + (0.973 + 0.230i)12-s + (1.06 − 1.43i)14-s + (0.597 + 0.802i)15-s + (−0.0581 + 0.998i)16-s + (−0.686 − 0.727i)18-s + (−0.686 + 0.727i)20-s + (−1.67 + 0.611i)21-s + ⋯
L(s)  = 1  + (0.396 − 0.918i)2-s + (−0.835 + 0.549i)3-s + (−0.686 − 0.727i)4-s + (−0.0581 − 0.998i)5-s + (0.173 + 0.984i)6-s + (1.73 + 0.412i)7-s + (−0.939 + 0.342i)8-s + (0.396 − 0.918i)9-s + (−0.939 − 0.342i)10-s + (0.973 + 0.230i)12-s + (1.06 − 1.43i)14-s + (0.597 + 0.802i)15-s + (−0.0581 + 0.998i)16-s + (−0.686 − 0.727i)18-s + (−0.686 + 0.727i)20-s + (−1.67 + 0.611i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.135 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.135 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $-0.135 + 0.990i$
Analytic conductor: \(0.808485\)
Root analytic conductor: \(0.899158\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (439, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :0),\ -0.135 + 0.990i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.127925625\)
\(L(\frac12)\) \(\approx\) \(1.127925625\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.396 + 0.918i)T \)
3 \( 1 + (0.835 - 0.549i)T \)
5 \( 1 + (0.0581 + 0.998i)T \)
good7 \( 1 + (-1.73 - 0.412i)T + (0.893 + 0.448i)T^{2} \)
11 \( 1 + (-0.396 - 0.918i)T^{2} \)
13 \( 1 + (-0.973 - 0.230i)T^{2} \)
17 \( 1 + (-0.173 + 0.984i)T^{2} \)
19 \( 1 + (-0.173 - 0.984i)T^{2} \)
23 \( 1 + (-1.89 + 0.448i)T + (0.893 - 0.448i)T^{2} \)
29 \( 1 + (0.342 + 0.460i)T + (-0.286 + 0.957i)T^{2} \)
31 \( 1 + (0.835 + 0.549i)T^{2} \)
37 \( 1 + (0.939 - 0.342i)T^{2} \)
41 \( 1 + (0.786 + 1.82i)T + (-0.686 + 0.727i)T^{2} \)
43 \( 1 + (1.67 - 0.843i)T + (0.597 - 0.802i)T^{2} \)
47 \( 1 + (0.227 + 0.758i)T + (-0.835 + 0.549i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (-0.396 + 0.918i)T^{2} \)
61 \( 1 + (-1.14 + 1.21i)T + (-0.0581 - 0.998i)T^{2} \)
67 \( 1 + (-0.713 + 0.957i)T + (-0.286 - 0.957i)T^{2} \)
71 \( 1 + (-0.766 - 0.642i)T^{2} \)
73 \( 1 + (-0.766 + 0.642i)T^{2} \)
79 \( 1 + (0.686 + 0.727i)T^{2} \)
83 \( 1 + (0.786 - 1.82i)T + (-0.686 - 0.727i)T^{2} \)
89 \( 1 + (0.744 - 0.270i)T + (0.766 - 0.642i)T^{2} \)
97 \( 1 + (0.993 + 0.116i)T^{2} \)
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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

−9.453479137823391413126405319244, −8.764061544680648368219430700688, −8.158359295075328109244931935510, −6.75529700874343176636579362244, −5.43934327741459856420811207824, −5.14591895166351334544434907567, −4.56934015367472458365648875357, −3.62535915129434509923119932629, −2.03642538658427038926008696549, −1.02135160696691827968826201653, 1.51176194053916016582421171641, 3.01970993263497830011118088401, 4.31601434424312537341902494598, 5.05843636073805213298963732958, 5.69002929756633432378763328863, 6.90189556667599775406955127060, 7.08090972825827576818845825958, 7.953990476013621339810674554243, 8.553662340534527100326601322646, 9.888293257632575061935354174601

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