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Properties

Label 2-585-9.4-c1-0-32
Degree $2$
Conductor $585$
Sign $0.835 + 0.549i$
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  + (0.238 + 0.413i)2-s + (1.59 − 0.686i)3-s + (0.886 − 1.53i)4-s + (−0.5 + 0.866i)5-s + (0.662 + 0.493i)6-s + (0.335 + 0.581i)7-s + 1.79·8-s + (2.05 − 2.18i)9-s − 0.477·10-s + (−1.85 − 3.20i)11-s + (0.355 − 3.04i)12-s + (0.5 − 0.866i)13-s + (−0.160 + 0.277i)14-s + (−0.200 + 1.72i)15-s + (−1.34 − 2.32i)16-s + 6.76·17-s + ⋯
L(s)  = 1  + (0.168 + 0.292i)2-s + (0.918 − 0.396i)3-s + (0.443 − 0.767i)4-s + (−0.223 + 0.387i)5-s + (0.270 + 0.201i)6-s + (0.126 + 0.219i)7-s + 0.636·8-s + (0.686 − 0.727i)9-s − 0.150·10-s + (−0.558 − 0.967i)11-s + (0.102 − 0.880i)12-s + (0.138 − 0.240i)13-s + (−0.0428 + 0.0741i)14-s + (−0.0518 + 0.444i)15-s + (−0.335 − 0.581i)16-s + 1.63·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.835 + 0.549i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.835 + 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $0.835 + 0.549i$
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: \((2,\ 585,\ (\ :1/2),\ 0.835 + 0.549i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.27040 - 0.679346i\)
\(L(\frac12)\) \(\approx\) \(2.27040 - 0.679346i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.59 + 0.686i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (-0.238 - 0.413i)T + (-1 + 1.73i)T^{2} \)
7 \( 1 + (-0.335 - 0.581i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.85 + 3.20i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 - 6.76T + 17T^{2} \)
19 \( 1 + 4.86T + 19T^{2} \)
23 \( 1 + (4.21 - 7.29i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.84 - 8.39i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.02 + 1.78i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 3.15T + 37T^{2} \)
41 \( 1 + (3.83 - 6.63i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.41 + 5.90i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.77 + 3.07i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 5.30T + 53T^{2} \)
59 \( 1 + (3.16 - 5.47i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.37 + 2.37i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.88 - 10.1i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 5.06T + 71T^{2} \)
73 \( 1 + 9.39T + 73T^{2} \)
79 \( 1 + (-7.13 - 12.3i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-8.34 - 14.4i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 4.23T + 89T^{2} \)
97 \( 1 + (0.471 + 0.816i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.41938465177288028283168705003, −9.915861272028492250017906599834, −8.623529016870990580910828930610, −7.923814246113630496965771245912, −7.10235955484328987604450856057, −6.10994180608199657106173891116, −5.29821877642746989746004518336, −3.70793278214432374751344576207, −2.71632760236198373048887765375, −1.34860591901103809377773037361, 1.94489307751297960158202369946, 2.94024435251596261962887188537, 4.15144184534592055972675477249, 4.66122050814034911454688530554, 6.39537599225656838134486867338, 7.69949168367750002022060327663, 7.952600265599827450580688325851, 8.902769430049272172377245965380, 10.17259829222481828776748834066, 10.46962661620107061516379238242

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