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Properties

Label 2-585-117.25-c1-0-8
Degree $2$
Conductor $585$
Sign $0.949 - 0.314i$
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  + (−0.883 − 0.509i)2-s + (−0.597 + 1.62i)3-s + (−0.480 − 0.831i)4-s + (−0.866 + 0.5i)5-s + (1.35 − 1.13i)6-s + (−3.32 − 1.91i)7-s + 3.01i·8-s + (−2.28 − 1.94i)9-s + 1.01·10-s + (−1.97 − 1.14i)11-s + (1.63 − 0.283i)12-s + (3.56 − 0.545i)13-s + (1.95 + 3.38i)14-s + (−0.295 − 1.70i)15-s + (0.578 − 1.00i)16-s + 7.50·17-s + ⋯
L(s)  = 1  + (−0.624 − 0.360i)2-s + (−0.344 + 0.938i)3-s + (−0.240 − 0.415i)4-s + (−0.387 + 0.223i)5-s + (0.553 − 0.461i)6-s + (−1.25 − 0.725i)7-s + 1.06i·8-s + (−0.762 − 0.647i)9-s + 0.322·10-s + (−0.596 − 0.344i)11-s + (0.473 − 0.0819i)12-s + (0.988 − 0.151i)13-s + (0.523 + 0.905i)14-s + (−0.0762 − 0.440i)15-s + (0.144 − 0.250i)16-s + 1.82·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.949 - 0.314i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.949 - 0.314i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $0.949 - 0.314i$
Analytic conductor: \(4.67124\)
Root analytic conductor: \(2.16130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{585} (376, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 585,\ (\ :1/2),\ 0.949 - 0.314i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.575839 + 0.0929575i\)
\(L(\frac12)\) \(\approx\) \(0.575839 + 0.0929575i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.597 - 1.62i)T \)
5 \( 1 + (0.866 - 0.5i)T \)
13 \( 1 + (-3.56 + 0.545i)T \)
good2 \( 1 + (0.883 + 0.509i)T + (1 + 1.73i)T^{2} \)
7 \( 1 + (3.32 + 1.91i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.97 + 1.14i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 - 7.50T + 17T^{2} \)
19 \( 1 - 2.31iT - 19T^{2} \)
23 \( 1 + (-1.68 - 2.91i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.07 - 7.06i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.49 - 2.01i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 0.546iT - 37T^{2} \)
41 \( 1 + (-6.35 + 3.67i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.826 - 1.43i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-7.13 - 4.11i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 - 7.10T + 53T^{2} \)
59 \( 1 + (-6.45 + 3.72i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.77 + 10.0i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.43 - 3.71i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 7.19iT - 71T^{2} \)
73 \( 1 - 7.74iT - 73T^{2} \)
79 \( 1 + (-6.19 + 10.7i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.81 - 3.35i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 6.46iT - 89T^{2} \)
97 \( 1 + (-10.1 - 5.83i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.64888703903421286620927917285, −9.967760907654123883787982422068, −9.322411840165098675279856608971, −8.388245987282979174431445739204, −7.28334550919106136535762182760, −5.91275638516258789933333710514, −5.37449585580749849420339172067, −3.80274068627446568435582322801, −3.21140374940184522585971071428, −0.844895523004378983565483584652, 0.65348570925770940751444196415, 2.69412644415228638070351886153, 3.81959696112375455982050424397, 5.48842641851092194245913292356, 6.28166004115476121182158496571, 7.27675127198569244402514990858, 7.909993127943311419686553582589, 8.780477709622335358567048565710, 9.523517485149779729284091212281, 10.54037822791000210720835903156

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