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Properties

Label 2-585-13.3-c1-0-14
Degree $2$
Conductor $585$
Sign $-0.964 + 0.265i$
Analytic cond. $4.67124$
Root an. cond. $2.16130$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.15 − 1.99i)2-s + (−1.65 + 2.86i)4-s + 5-s + (0.5 − 0.866i)7-s + 2.99·8-s + (−1.15 − 1.99i)10-s + (0.802 + 1.39i)11-s − 3.60·13-s − 2.30·14-s + (−0.151 − 0.262i)16-s + (3.80 − 6.58i)17-s + (2.80 − 4.85i)19-s + (−1.65 + 2.86i)20-s + (1.84 − 3.20i)22-s + (−1.5 − 2.59i)23-s + ⋯
L(s)  = 1  + (−0.814 − 1.41i)2-s + (−0.825 + 1.43i)4-s + 0.447·5-s + (0.188 − 0.327i)7-s + 1.06·8-s + (−0.364 − 0.630i)10-s + (0.242 + 0.419i)11-s − 1.00·13-s − 0.615·14-s + (−0.0378 − 0.0655i)16-s + (0.922 − 1.59i)17-s + (0.643 − 1.11i)19-s + (−0.369 + 0.639i)20-s + (0.394 − 0.682i)22-s + (−0.312 − 0.541i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.964 + 0.265i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.964 + 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $-0.964 + 0.265i$
Analytic conductor: \(4.67124\)
Root analytic conductor: \(2.16130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{585} (406, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 585,\ (\ :1/2),\ -0.964 + 0.265i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.112808 - 0.836139i\)
\(L(\frac12)\) \(\approx\) \(0.112808 - 0.836139i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 - T \)
13 \( 1 + 3.60T \)
good2 \( 1 + (1.15 + 1.99i)T + (-1 + 1.73i)T^{2} \)
7 \( 1 + (-0.5 + 0.866i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.802 - 1.39i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-3.80 + 6.58i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.80 + 4.85i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.5 + 2.59i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.10 + 5.37i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + (1.80 + 3.12i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.5 - 2.59i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.10 + 8.84i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 9.21T + 47T^{2} \)
53 \( 1 - 3.21T + 53T^{2} \)
59 \( 1 + (5.40 - 9.36i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.5 - 6.06i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-2.40 + 4.17i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 0.788T + 73T^{2} \)
79 \( 1 - 5.21T + 79T^{2} \)
83 \( 1 - 9.21T + 83T^{2} \)
89 \( 1 + (3.10 + 5.37i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-4.19 + 7.26i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.21140043790236143054577420641, −9.529776251069142339418033751739, −9.113343386017076619365699633878, −7.75921891865245988950491983430, −7.07878304512902982558008474063, −5.46786814581091082008414050134, −4.37578571209581520864890986887, −3.01556184242093139969566683202, −2.13491309999883799829587654311, −0.64903909413762641751616505569, 1.57469186591340619258495620963, 3.50539217898122236647920480363, 5.22083801248646958759740155185, 5.77434285912283199724715768955, 6.65219663898547700268366944517, 7.74095564126832266571094937208, 8.202558594466820192843259205579, 9.280584799600831065390738957775, 9.829365792666168927700243919955, 10.71459207430285839300473948724

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