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Properties

Label 2-585-117.25-c1-0-14
Degree $2$
Conductor $585$
Sign $-0.195 + 0.980i$
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  + (−2.24 − 1.29i)2-s + (−0.927 + 1.46i)3-s + (2.35 + 4.07i)4-s + (−0.866 + 0.5i)5-s + (3.97 − 2.08i)6-s + (−1.32 − 0.766i)7-s − 7.00i·8-s + (−1.28 − 2.71i)9-s + 2.58·10-s + (0.217 + 0.125i)11-s + (−8.14 − 0.336i)12-s + (0.215 + 3.59i)13-s + (1.98 + 3.43i)14-s + (0.0714 − 1.73i)15-s + (−4.35 + 7.54i)16-s − 5.82·17-s + ⋯
L(s)  = 1  + (−1.58 − 0.915i)2-s + (−0.535 + 0.844i)3-s + (1.17 + 2.03i)4-s + (−0.387 + 0.223i)5-s + (1.62 − 0.849i)6-s + (−0.501 − 0.289i)7-s − 2.47i·8-s + (−0.426 − 0.904i)9-s + 0.818·10-s + (0.0655 + 0.0378i)11-s + (−2.34 − 0.0970i)12-s + (0.0598 + 0.998i)13-s + (0.530 + 0.918i)14-s + (0.0184 − 0.446i)15-s + (−1.08 + 1.88i)16-s − 1.41·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.195 + 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.195 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $-0.195 + 0.980i$
Analytic conductor: \(4.67124\)
Root analytic conductor: \(2.16130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{585} (376, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 585,\ (\ :1/2),\ -0.195 + 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.143380 - 0.174847i\)
\(L(\frac12)\) \(\approx\) \(0.143380 - 0.174847i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.927 - 1.46i)T \)
5 \( 1 + (0.866 - 0.5i)T \)
13 \( 1 + (-0.215 - 3.59i)T \)
good2 \( 1 + (2.24 + 1.29i)T + (1 + 1.73i)T^{2} \)
7 \( 1 + (1.32 + 0.766i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.217 - 0.125i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + 5.82T + 17T^{2} \)
19 \( 1 - 4.62iT - 19T^{2} \)
23 \( 1 + (1.46 + 2.53i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.12 + 3.68i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.0701 + 0.0404i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 3.43iT - 37T^{2} \)
41 \( 1 + (-7.46 + 4.30i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.97 + 5.14i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.97 + 2.29i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 - 5.86T + 53T^{2} \)
59 \( 1 + (7.40 - 4.27i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.243 - 0.421i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.56 + 2.05i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 7.73iT - 71T^{2} \)
73 \( 1 + 14.7iT - 73T^{2} \)
79 \( 1 + (-6.91 + 11.9i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-11.8 - 6.82i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 5.99iT - 89T^{2} \)
97 \( 1 + (7.77 + 4.48i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.64055601268923851667825911315, −9.580392833563992889371110811300, −9.094858745041360720988676415128, −8.177455207634916477065697819346, −7.03366693864358104788119509813, −6.23663920827649098639717294944, −4.34069113174015827294344372856, −3.57464282232793491232663869566, −2.18737174385606543577017947083, −0.28319308689456851703634766918, 1.00286464345550667192686610557, 2.58070386318723207311319221944, 4.92956224428608244560868097873, 6.00644934285855505947918540893, 6.65977720943408363417683777601, 7.46206448497793769313995329167, 8.236895879962324041582504260008, 8.945397055590872924330280678539, 9.843349753549029121730352022148, 10.95730503622791526433708799025

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