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Properties

Label 2-585-9.4-c1-0-45
Degree $2$
Conductor $585$
Sign $0.284 - 0.958i$
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  + (−1.33 − 2.30i)2-s + (0.397 − 1.68i)3-s + (−2.54 + 4.40i)4-s + (0.5 − 0.866i)5-s + (−4.41 + 1.32i)6-s + (−0.723 − 1.25i)7-s + 8.22·8-s + (−2.68 − 1.34i)9-s − 2.66·10-s + (−3.18 − 5.51i)11-s + (6.41 + 6.04i)12-s + (−0.5 + 0.866i)13-s + (−1.92 + 3.33i)14-s + (−1.26 − 1.18i)15-s + (−5.85 − 10.1i)16-s + 3.28·17-s + ⋯
L(s)  = 1  + (−0.941 − 1.63i)2-s + (0.229 − 0.973i)3-s + (−1.27 + 2.20i)4-s + (0.223 − 0.387i)5-s + (−1.80 + 0.541i)6-s + (−0.273 − 0.473i)7-s + 2.90·8-s + (−0.894 − 0.446i)9-s − 0.841·10-s + (−0.959 − 1.66i)11-s + (1.85 + 1.74i)12-s + (−0.138 + 0.240i)13-s + (−0.514 + 0.891i)14-s + (−0.325 − 0.306i)15-s + (−1.46 − 2.53i)16-s + 0.797·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.284 - 0.958i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.284 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $0.284 - 0.958i$
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: \((2,\ 585,\ (\ :1/2),\ 0.284 - 0.958i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.392775 + 0.293114i\)
\(L(\frac12)\) \(\approx\) \(0.392775 + 0.293114i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.397 + 1.68i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (1.33 + 2.30i)T + (-1 + 1.73i)T^{2} \)
7 \( 1 + (0.723 + 1.25i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (3.18 + 5.51i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 - 3.28T + 17T^{2} \)
19 \( 1 + 2.89T + 19T^{2} \)
23 \( 1 + (-0.613 + 1.06i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.73 - 8.20i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.20 - 2.08i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 4.81T + 37T^{2} \)
41 \( 1 + (2.75 - 4.77i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.60 - 6.24i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.00 + 6.93i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 0.623T + 53T^{2} \)
59 \( 1 + (-4.93 + 8.55i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.24 + 3.88i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.63 + 2.82i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 9.19T + 71T^{2} \)
73 \( 1 - 9.55T + 73T^{2} \)
79 \( 1 + (7.82 + 13.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.526 - 0.911i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 2.96T + 89T^{2} \)
97 \( 1 + (-5.76 - 9.99i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.21776449130217465444863662987, −9.071095806316538194758248788467, −8.439753367915034478112092914865, −7.87495180836221416866486066411, −6.64155857590913126521941469446, −5.19126652853948179884469411770, −3.50949328833662391108055013914, −2.83754714555895636909729542633, −1.49830497678016125634671969876, −0.37275849673726745289202299225, 2.41777783827326841116659126262, 4.33098028067215388715330527823, 5.27991832393248712391685388293, 5.96622441926432329079680241744, 7.13874547596973673036532668713, 7.85941983588068902808501120839, 8.709361255028377936546892272080, 9.661574472586989920239600371243, 10.04217270324947109694013871580, 10.68359587905958002488346764339

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