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Properties

Label 2-585-9.7-c1-0-1
Degree $2$
Conductor $585$
Sign $-0.425 + 0.905i$
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  + (−1.22 + 2.12i)2-s + (−1.71 − 0.228i)3-s + (−1.99 − 3.45i)4-s + (0.5 + 0.866i)5-s + (2.58 − 3.36i)6-s + (1.96 − 3.40i)7-s + 4.88·8-s + (2.89 + 0.784i)9-s − 2.44·10-s + (−2.33 + 4.04i)11-s + (2.63 + 6.39i)12-s + (0.5 + 0.866i)13-s + (4.81 + 8.34i)14-s + (−0.660 − 1.60i)15-s + (−1.98 + 3.43i)16-s − 7.96·17-s + ⋯
L(s)  = 1  + (−0.865 + 1.49i)2-s + (−0.991 − 0.131i)3-s + (−0.998 − 1.72i)4-s + (0.223 + 0.387i)5-s + (1.05 − 1.37i)6-s + (0.743 − 1.28i)7-s + 1.72·8-s + (0.965 + 0.261i)9-s − 0.774·10-s + (−0.704 + 1.22i)11-s + (0.761 + 1.84i)12-s + (0.138 + 0.240i)13-s + (1.28 + 2.22i)14-s + (−0.170 − 0.413i)15-s + (−0.495 + 0.858i)16-s − 1.93·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.425 + 0.905i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.425 + 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $-0.425 + 0.905i$
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: \((2,\ 585,\ (\ :1/2),\ -0.425 + 0.905i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0695993 - 0.109573i\)
\(L(\frac12)\) \(\approx\) \(0.0695993 - 0.109573i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.71 + 0.228i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (-0.5 - 0.866i)T \)
good2 \( 1 + (1.22 - 2.12i)T + (-1 - 1.73i)T^{2} \)
7 \( 1 + (-1.96 + 3.40i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.33 - 4.04i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + 7.96T + 17T^{2} \)
19 \( 1 + 0.120T + 19T^{2} \)
23 \( 1 + (0.164 + 0.284i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.136 + 0.236i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.28 - 5.69i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 7.43T + 37T^{2} \)
41 \( 1 + (0.0455 + 0.0788i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.89 - 6.75i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.78 + 10.0i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 11.1T + 53T^{2} \)
59 \( 1 + (2.77 + 4.81i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.63 - 11.4i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.41 + 5.90i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 6.32T + 71T^{2} \)
73 \( 1 + 7.37T + 73T^{2} \)
79 \( 1 + (-5.95 + 10.3i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.774 - 1.34i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 3.59T + 89T^{2} \)
97 \( 1 + (2.63 - 4.57i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.68161576479550778844807752016, −10.53351157521946458468975790416, −9.528559935535686918978650739963, −8.386486086512732800867318986734, −7.36914608131409111749147972065, −6.99741103920847296049086853832, −6.27478195199420452738566390663, −4.94919985313441949329677127178, −4.49689393838767806751401622992, −1.64884253743066931481331033992, 0.11133264065886383651619888157, 1.69305604000094333125267356719, 2.77391201707971928547022100710, 4.31332992313106294758834331250, 5.33269963575003673882393912929, 6.24354695114939431446216780113, 7.948512514550541430771809685528, 8.739990914526727155855018996626, 9.262128413283976289918944186797, 10.42408203605097420891711877039

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