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Properties

Label 2-585-9.7-c1-0-9
Degree $2$
Conductor $585$
Sign $0.913 - 0.407i$
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  + (0.768 − 1.33i)2-s + (−1.52 − 0.812i)3-s + (−0.181 − 0.314i)4-s + (−0.5 − 0.866i)5-s + (−2.25 + 1.41i)6-s + (−1.69 + 2.94i)7-s + 2.51·8-s + (1.68 + 2.48i)9-s − 1.53·10-s + (−3.21 + 5.56i)11-s + (0.0222 + 0.628i)12-s + (0.5 + 0.866i)13-s + (2.61 + 4.52i)14-s + (0.0613 + 1.73i)15-s + (2.29 − 3.97i)16-s + 2.03·17-s + ⋯
L(s)  = 1  + (0.543 − 0.941i)2-s + (−0.883 − 0.468i)3-s + (−0.0907 − 0.157i)4-s + (−0.223 − 0.387i)5-s + (−0.921 + 0.576i)6-s + (−0.642 + 1.11i)7-s + 0.889·8-s + (0.560 + 0.828i)9-s − 0.486·10-s + (−0.967 + 1.67i)11-s + (0.00643 + 0.181i)12-s + (0.138 + 0.240i)13-s + (0.698 + 1.20i)14-s + (0.0158 + 0.446i)15-s + (0.574 − 0.994i)16-s + 0.493·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.913 - 0.407i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.913 - 0.407i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $0.913 - 0.407i$
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: \((2,\ 585,\ (\ :1/2),\ 0.913 - 0.407i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.06871 + 0.227769i\)
\(L(\frac12)\) \(\approx\) \(1.06871 + 0.227769i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.52 + 0.812i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (-0.5 - 0.866i)T \)
good2 \( 1 + (-0.768 + 1.33i)T + (-1 - 1.73i)T^{2} \)
7 \( 1 + (1.69 - 2.94i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (3.21 - 5.56i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 - 2.03T + 17T^{2} \)
19 \( 1 + 4.38T + 19T^{2} \)
23 \( 1 + (-3.61 - 6.25i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.38 - 2.39i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (5.44 + 9.42i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 6.69T + 37T^{2} \)
41 \( 1 + (-3.54 - 6.14i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.74 - 6.48i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.88 + 4.99i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 1.37T + 53T^{2} \)
59 \( 1 + (-0.514 - 0.891i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.82 - 11.8i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.72 + 8.17i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 1.12T + 71T^{2} \)
73 \( 1 - 4.90T + 73T^{2} \)
79 \( 1 + (-0.277 + 0.479i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (6.32 - 10.9i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 6.37T + 89T^{2} \)
97 \( 1 + (-7.25 + 12.5i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.11154534570048845095702165333, −10.08709026527626947834229753511, −9.359085691381047810212157999563, −7.84803854594001692405940659924, −7.24820714957829774607728876738, −5.94911963360461350043652255732, −5.07336839402994732567716324626, −4.21306499107762336688581581328, −2.67538282908813838611137034523, −1.74973628233529977231872721292, 0.57257037022373310787005193885, 3.27790619762699337810566907637, 4.22764051614761391252661912746, 5.30514435253894982898404412418, 6.09647362777163174571467862947, 6.76814064562308354596376373617, 7.60671109264914668799259596684, 8.716462310994256071732096016562, 10.23268140004549060924762477668, 10.70636287283551680419254386568

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