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Properties

Label 2-585-9.4-c1-0-1
Degree $2$
Conductor $585$
Sign $-0.255 - 0.966i$
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  + (−0.877 − 1.51i)2-s + (0.0726 + 1.73i)3-s + (−0.540 + 0.935i)4-s + (−0.5 + 0.866i)5-s + (2.56 − 1.62i)6-s + (1.83 + 3.17i)7-s − 1.61·8-s + (−2.98 + 0.251i)9-s + 1.75·10-s + (−0.687 − 1.19i)11-s + (−1.65 − 0.866i)12-s + (0.5 − 0.866i)13-s + (3.22 − 5.57i)14-s + (−1.53 − 0.802i)15-s + (2.49 + 4.32i)16-s − 0.562·17-s + ⋯
L(s)  = 1  + (−0.620 − 1.07i)2-s + (0.0419 + 0.999i)3-s + (−0.270 + 0.467i)4-s + (−0.223 + 0.387i)5-s + (1.04 − 0.665i)6-s + (0.693 + 1.20i)7-s − 0.570·8-s + (−0.996 + 0.0837i)9-s + 0.555·10-s + (−0.207 − 0.359i)11-s + (−0.478 − 0.250i)12-s + (0.138 − 0.240i)13-s + (0.860 − 1.49i)14-s + (−0.396 − 0.207i)15-s + (0.624 + 1.08i)16-s − 0.136·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.255 - 0.966i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.255 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $-0.255 - 0.966i$
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: \((2,\ 585,\ (\ :1/2),\ -0.255 - 0.966i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.342254 + 0.444469i\)
\(L(\frac12)\) \(\approx\) \(0.342254 + 0.444469i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.0726 - 1.73i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (0.877 + 1.51i)T + (-1 + 1.73i)T^{2} \)
7 \( 1 + (-1.83 - 3.17i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.687 + 1.19i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + 0.562T + 17T^{2} \)
19 \( 1 + 4.54T + 19T^{2} \)
23 \( 1 + (3.65 - 6.33i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.40 + 7.63i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.02 - 5.23i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 0.137T + 37T^{2} \)
41 \( 1 + (4.13 - 7.17i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.17 - 2.03i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.710 - 1.23i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 5.26T + 53T^{2} \)
59 \( 1 + (4.53 - 7.84i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.50 - 4.33i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.13 - 5.43i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 3.03T + 71T^{2} \)
73 \( 1 - 8.25T + 73T^{2} \)
79 \( 1 + (2.13 + 3.70i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.119 - 0.206i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 2.59T + 89T^{2} \)
97 \( 1 + (-6.05 - 10.4i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.93504836182579954313244284634, −10.18664666336795289479721237407, −9.361533789306394575686856831907, −8.643542415688846118405130128448, −7.959260478364316005510147006830, −6.10627738752590855228490139477, −5.42428705208123045786233594132, −4.03866795485589159012565631761, −2.97198108426759271334569460583, −2.01412694996488150902607262753, 0.36289902787983826366084617800, 2.00395141908017386981018219652, 3.85233452001191472980754732777, 5.13946620478324794258365878032, 6.32847056901908774478464662998, 7.06633326929981406559373599603, 7.70456041683706134508611883894, 8.383655312914102089820271570361, 9.078898789760690294088650783247, 10.43820901232407999791929920165

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