↑

Properties

Label 2-585-9.4-c1-0-46
Degree $2$
Conductor $585$
Sign $-0.863 - 0.504i$
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  + (−0.332 − 0.575i)2-s + (−0.727 − 1.57i)3-s + (0.779 − 1.34i)4-s + (0.5 − 0.866i)5-s + (−0.662 + 0.940i)6-s + (−1.40 − 2.43i)7-s − 2.36·8-s + (−1.94 + 2.28i)9-s − 0.664·10-s + (−0.115 − 0.199i)11-s + (−2.68 − 0.243i)12-s + (−0.5 + 0.866i)13-s + (−0.933 + 1.61i)14-s + (−1.72 − 0.155i)15-s + (−0.772 − 1.33i)16-s − 3.24·17-s + ⋯
L(s)  = 1  + (−0.234 − 0.406i)2-s + (−0.420 − 0.907i)3-s + (0.389 − 0.674i)4-s + (0.223 − 0.387i)5-s + (−0.270 + 0.384i)6-s + (−0.530 − 0.919i)7-s − 0.835·8-s + (−0.647 + 0.762i)9-s − 0.210·10-s + (−0.0347 − 0.0601i)11-s + (−0.776 − 0.0701i)12-s + (−0.138 + 0.240i)13-s + (−0.249 + 0.432i)14-s + (−0.445 − 0.0402i)15-s + (−0.193 − 0.334i)16-s − 0.786·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.863 - 0.504i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.863 - 0.504i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $-0.863 - 0.504i$
Analytic conductor: \(4.67124\)
Root analytic conductor: \(2.16130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{585} (391, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 585,\ (\ :1/2),\ -0.863 - 0.504i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.218588 + 0.806467i\)
\(L(\frac12)\) \(\approx\) \(0.218588 + 0.806467i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.727 + 1.57i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (0.332 + 0.575i)T + (-1 + 1.73i)T^{2} \)
7 \( 1 + (1.40 + 2.43i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.115 + 0.199i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + 3.24T + 17T^{2} \)
19 \( 1 - 3.80T + 19T^{2} \)
23 \( 1 + (-1.25 + 2.16i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.64 - 4.58i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.05 + 3.56i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 2.95T + 37T^{2} \)
41 \( 1 + (1.74 - 3.02i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.66 + 9.81i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.720 - 1.24i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 3.34T + 53T^{2} \)
59 \( 1 + (3.89 - 6.74i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.04 + 8.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.00 - 1.74i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 3.51T + 71T^{2} \)
73 \( 1 + 5.30T + 73T^{2} \)
79 \( 1 + (-2.69 - 4.67i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.0501 + 0.0869i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 7.04T + 89T^{2} \)
97 \( 1 + (1.63 + 2.82i)T + (-48.5 + 84.0i)T^{2} \)
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

−10.33649726756338805953922571899, −9.487728503287296611837371555244, −8.494344595459180558525221357185, −7.23330004353133800361522378807, −6.65160436081191767547217292599, −5.76099460893053761499526726502, −4.69262090085807985409961275345, −2.97388839393007362837397995277, −1.68607171589086736753117834422, −0.51620377898790969442467682247, 2.64847584646766060424726471280, 3.40664288130763905715277106059, 4.81614152162617374638438403389, 5.96494840966091368181236010302, 6.51499842172776284099746669486, 7.66929470533959900647213621499, 8.754190093850943740251649979065, 9.388085273977781358516049469493, 10.22065174958061457173365957237, 11.30244646132320058562153308387

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