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Properties

Label 2-585-9.4-c1-0-17
Degree $2$
Conductor $585$
Sign $-0.512 + 0.858i$
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  + (−1.12 − 1.95i)2-s + (−0.604 + 1.62i)3-s + (−1.54 + 2.67i)4-s + (0.5 − 0.866i)5-s + (3.85 − 0.650i)6-s + (−0.353 − 0.611i)7-s + 2.46·8-s + (−2.26 − 1.96i)9-s − 2.25·10-s + (1.16 + 2.01i)11-s + (−3.41 − 4.12i)12-s + (0.5 − 0.866i)13-s + (−0.797 + 1.38i)14-s + (1.10 + 1.33i)15-s + (0.311 + 0.538i)16-s + 6.04·17-s + ⋯
L(s)  = 1  + (−0.797 − 1.38i)2-s + (−0.348 + 0.937i)3-s + (−0.773 + 1.33i)4-s + (0.223 − 0.387i)5-s + (1.57 − 0.265i)6-s + (−0.133 − 0.231i)7-s + 0.871·8-s + (−0.756 − 0.653i)9-s − 0.713·10-s + (0.351 + 0.608i)11-s + (−0.985 − 1.19i)12-s + (0.138 − 0.240i)13-s + (−0.213 + 0.369i)14-s + (0.284 + 0.344i)15-s + (0.0777 + 0.134i)16-s + 1.46·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.512 + 0.858i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.512 + 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $-0.512 + 0.858i$
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: \((2,\ 585,\ (\ :1/2),\ -0.512 + 0.858i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.345739 - 0.609008i\)
\(L(\frac12)\) \(\approx\) \(0.345739 - 0.609008i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.604 - 1.62i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (1.12 + 1.95i)T + (-1 + 1.73i)T^{2} \)
7 \( 1 + (0.353 + 0.611i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.16 - 2.01i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 - 6.04T + 17T^{2} \)
19 \( 1 + 7.91T + 19T^{2} \)
23 \( 1 + (-4.47 + 7.75i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.20 - 2.08i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.53 + 2.65i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 2.38T + 37T^{2} \)
41 \( 1 + (-4.72 + 8.19i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.96 + 10.3i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.86 + 8.42i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 5.90T + 53T^{2} \)
59 \( 1 + (1.48 - 2.57i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.62 - 8.00i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.847 + 1.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 12.4T + 71T^{2} \)
73 \( 1 - 4.88T + 73T^{2} \)
79 \( 1 + (-5.51 - 9.54i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.260 - 0.451i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 10.7T + 89T^{2} \)
97 \( 1 + (-1.48 - 2.56i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.33762618041556527524372562054, −9.930100225710811139823003866235, −8.799879338158267697461138780561, −8.509449420474346899040103307878, −6.83042885583080584733014337468, −5.58092308617777680714362120010, −4.39107240063461259796809324578, −3.54759445418766346876414119286, −2.25645023095001832941158901293, −0.61248553333314483788196496609, 1.27020575592909544872021576099, 3.08077349818396808347052816465, 5.07759458135897438763842630308, 6.16931871912041146567709209947, 6.36118768891270130901552449234, 7.48093430842563305060032553412, 8.087095334110826829716642042152, 8.948301586931863134976871967156, 9.826485762734117642977226124136, 10.92163301196826687927371033980

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