Properties

Label 2-585-9.7-c1-0-34
Degree $2$
Conductor $585$
Sign $0.766 + 0.642i$
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  + (0.5 − 0.866i)2-s + (1.5 − 0.866i)3-s + (0.500 + 0.866i)4-s + (0.5 + 0.866i)5-s − 1.73i·6-s + (0.5 − 0.866i)7-s + 3·8-s + (1.5 − 2.59i)9-s + 0.999·10-s + (−1 + 1.73i)11-s + (1.5 + 0.866i)12-s + (0.5 + 0.866i)13-s + (−0.499 − 0.866i)14-s + (1.5 + 0.866i)15-s + (0.500 − 0.866i)16-s − 4·17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.866 − 0.499i)3-s + (0.250 + 0.433i)4-s + (0.223 + 0.387i)5-s − 0.707i·6-s + (0.188 − 0.327i)7-s + 1.06·8-s + (0.5 − 0.866i)9-s + 0.316·10-s + (−0.301 + 0.522i)11-s + (0.433 + 0.249i)12-s + (0.138 + 0.240i)13-s + (−0.133 − 0.231i)14-s + (0.387 + 0.223i)15-s + (0.125 − 0.216i)16-s − 0.970·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $0.766 + 0.642i$
Analytic conductor: \(4.67124\)
Root analytic conductor: \(2.16130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{585} (196, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 585,\ (\ :1/2),\ 0.766 + 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.53444 - 0.922464i\)
\(L(\frac12)\) \(\approx\) \(2.53444 - 0.922464i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.5 + 0.866i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (-0.5 - 0.866i)T \)
good2 \( 1 + (-0.5 + 0.866i)T + (-1 - 1.73i)T^{2} \)
7 \( 1 + (-0.5 + 0.866i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1 - 1.73i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + 4T + 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + (-1.5 - 2.59i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (4 + 6.92i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 4T + 37T^{2} \)
41 \( 1 + (4.5 + 7.79i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4 + 6.92i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (6.5 - 11.2i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 10T + 53T^{2} \)
59 \( 1 + (3 + 5.19i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + 12T + 73T^{2} \)
79 \( 1 + (-3 + 5.19i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.5 - 9.52i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 5T + 89T^{2} \)
97 \( 1 + (-1 + 1.73i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.79821951436310820075465847427, −9.762232686301128468934122640461, −8.872052845965269038322760928171, −7.71580590381207353225188753825, −7.29581762595020488631745951599, −6.26834258761437376344374484753, −4.57754782812292633278420179170, −3.68305172604854356812886061121, −2.60085713749009244616324541669, −1.75268678312710092264909982369, 1.72194128531540891089349239436, 3.01130158522436607329228570294, 4.48588216180960715300826204209, 5.14691771028516477616526773907, 6.18656204873664778335274499982, 7.20963405022649932066197688874, 8.267573270072988521579491972928, 8.865223793065681454817506951698, 9.905156020513125021216327826993, 10.66533307081293798832379055621

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