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Properties

Label 2-585-9.4-c1-0-21
Degree $2$
Conductor $585$
Sign $0.985 + 0.168i$
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  + (−0.327 − 0.567i)2-s + (0.997 + 1.41i)3-s + (0.785 − 1.36i)4-s + (0.5 − 0.866i)5-s + (0.476 − 1.02i)6-s + (0.388 + 0.673i)7-s − 2.33·8-s + (−1.01 + 2.82i)9-s − 0.655·10-s + (2.03 + 3.52i)11-s + (2.70 − 0.244i)12-s + (0.5 − 0.866i)13-s + (0.254 − 0.440i)14-s + (1.72 − 0.155i)15-s + (−0.804 − 1.39i)16-s + 5.29·17-s + ⋯
L(s)  = 1  + (−0.231 − 0.401i)2-s + (0.575 + 0.817i)3-s + (0.392 − 0.680i)4-s + (0.223 − 0.387i)5-s + (0.194 − 0.420i)6-s + (0.146 + 0.254i)7-s − 0.827·8-s + (−0.337 + 0.941i)9-s − 0.207·10-s + (0.613 + 1.06i)11-s + (0.782 − 0.0704i)12-s + (0.138 − 0.240i)13-s + (0.0680 − 0.117i)14-s + (0.445 − 0.0401i)15-s + (−0.201 − 0.348i)16-s + 1.28·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.168i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 + 0.168i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $0.985 + 0.168i$
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: \((2,\ 585,\ (\ :1/2),\ 0.985 + 0.168i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.85717 - 0.157660i\)
\(L(\frac12)\) \(\approx\) \(1.85717 - 0.157660i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.997 - 1.41i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (0.327 + 0.567i)T + (-1 + 1.73i)T^{2} \)
7 \( 1 + (-0.388 - 0.673i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.03 - 3.52i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 - 5.29T + 17T^{2} \)
19 \( 1 - 6.65T + 19T^{2} \)
23 \( 1 + (1.15 - 1.99i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.19 + 5.52i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.47 + 7.74i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 2.80T + 37T^{2} \)
41 \( 1 + (3.80 - 6.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.42 + 7.65i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.04 - 3.54i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 4.78T + 53T^{2} \)
59 \( 1 + (-2.37 + 4.10i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7.18 - 12.4i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.39 - 9.34i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 0.307T + 71T^{2} \)
73 \( 1 + 7.50T + 73T^{2} \)
79 \( 1 + (3.80 + 6.59i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-6.26 - 10.8i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 11.7T + 89T^{2} \)
97 \( 1 + (-3.04 - 5.27i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.30046525151648708163193573768, −9.768083441573050302345825849144, −9.386474003772672090349886479926, −8.220971123329218416053886758789, −7.26370980185038725211035254019, −5.80365117976328234397103315039, −5.17644845327777031588568746888, −3.90817536965538627372511970006, −2.67141626228240150485758561019, −1.46609976303041572358017334039, 1.37595211080371477162542160750, 3.06168391522810254675223501588, 3.50760815578544907697395060118, 5.55827945872431379559704301607, 6.53933769734545670732847322263, 7.18622611285354470179050130131, 7.999240408675441746105078934543, 8.711326820384575597728632555723, 9.566144245625156027828917665274, 10.82504342839047041546153991827

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