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Properties

Label 2-1620-1620.1579-c0-0-1
Degree $2$
Conductor $1620$
Sign $0.952 - 0.305i$
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  + (0.0581 + 0.998i)2-s + (0.686 − 0.727i)3-s + (−0.993 + 0.116i)4-s + (0.973 + 0.230i)5-s + (0.766 + 0.642i)6-s + (0.342 − 0.460i)7-s + (−0.173 − 0.984i)8-s + (−0.0581 − 0.998i)9-s + (−0.173 + 0.984i)10-s + (−0.597 + 0.802i)12-s + (0.479 + 0.315i)14-s + (0.835 − 0.549i)15-s + (0.973 − 0.230i)16-s + (0.993 − 0.116i)18-s + (−0.993 − 0.116i)20-s + (−0.0996 − 0.564i)21-s + ⋯
L(s)  = 1  + (0.0581 + 0.998i)2-s + (0.686 − 0.727i)3-s + (−0.993 + 0.116i)4-s + (0.973 + 0.230i)5-s + (0.766 + 0.642i)6-s + (0.342 − 0.460i)7-s + (−0.173 − 0.984i)8-s + (−0.0581 − 0.998i)9-s + (−0.173 + 0.984i)10-s + (−0.597 + 0.802i)12-s + (0.479 + 0.315i)14-s + (0.835 − 0.549i)15-s + (0.973 − 0.230i)16-s + (0.993 − 0.116i)18-s + (−0.993 − 0.116i)20-s + (−0.0996 − 0.564i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 - 0.305i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 - 0.305i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $0.952 - 0.305i$
Analytic conductor: \(0.808485\)
Root analytic conductor: \(0.899158\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (1579, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :0),\ 0.952 - 0.305i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.544124937\)
\(L(\frac12)\) \(\approx\) \(1.544124937\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0581 - 0.998i)T \)
3 \( 1 + (-0.686 + 0.727i)T \)
5 \( 1 + (-0.973 - 0.230i)T \)
good7 \( 1 + (-0.342 + 0.460i)T + (-0.286 - 0.957i)T^{2} \)
11 \( 1 + (0.0581 - 0.998i)T^{2} \)
13 \( 1 + (-0.597 + 0.802i)T^{2} \)
17 \( 1 + (-0.766 + 0.642i)T^{2} \)
19 \( 1 + (-0.766 - 0.642i)T^{2} \)
23 \( 1 + (0.713 + 0.957i)T + (-0.286 + 0.957i)T^{2} \)
29 \( 1 + (0.661 - 0.435i)T + (0.396 - 0.918i)T^{2} \)
31 \( 1 + (0.686 + 0.727i)T^{2} \)
37 \( 1 + (-0.173 - 0.984i)T^{2} \)
41 \( 1 + (0.103 - 1.78i)T + (-0.993 - 0.116i)T^{2} \)
43 \( 1 + (-0.0996 + 0.332i)T + (-0.835 - 0.549i)T^{2} \)
47 \( 1 + (-0.0460 - 0.106i)T + (-0.686 + 0.727i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.0581 + 0.998i)T^{2} \)
61 \( 1 + (-1.36 - 0.159i)T + (0.973 + 0.230i)T^{2} \)
67 \( 1 + (1.39 + 0.918i)T + (0.396 + 0.918i)T^{2} \)
71 \( 1 + (0.939 + 0.342i)T^{2} \)
73 \( 1 + (0.939 - 0.342i)T^{2} \)
79 \( 1 + (0.993 - 0.116i)T^{2} \)
83 \( 1 + (-0.103 - 1.78i)T + (-0.993 + 0.116i)T^{2} \)
89 \( 1 + (0.0201 + 0.114i)T + (-0.939 + 0.342i)T^{2} \)
97 \( 1 + (-0.893 + 0.448i)T^{2} \)
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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

−9.432028274862194501205521177822, −8.642741761071790111076741663653, −7.959874582089358629206495749964, −7.19447800385805553040227411427, −6.49490156402350547371960164721, −5.86432410343698509931837081308, −4.80292077832226152390921494133, −3.76311700061395016995097711674, −2.60917343934653932576518961458, −1.32216659735575445032379766997, 1.74618670107951813089117307973, 2.41140200584990616556684099645, 3.48898855373383949963907384738, 4.37845205152473786712146915465, 5.33035658443645678038978088170, 5.80416325103176669635170189720, 7.42470296048376818473162469933, 8.463731577029314529076869190647, 8.915161273983658695543497588297, 9.698920104603022454681089687800

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