↑

Properties

Label 2-585-13.9-c1-0-2
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

Related objects

Downloads

Learn more

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: $\chi_{585} (451, \cdot )$
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 \( 1 + (-4.19 - 7.26i)T + (-48.5 + 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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