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Properties

Label 1-585-585.308-r0-0-0
Degree $1$
Conductor $585$
Sign $-0.839 - 0.543i$
Analytic cond. $2.71672$
Root an. cond. $2.71672$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 − 0.5i)2-s + (0.5 + 0.866i)4-s − i·7-s − i·8-s + (0.5 − 0.866i)11-s + (−0.5 + 0.866i)14-s + (−0.5 + 0.866i)16-s + (−0.866 − 0.5i)17-s + (0.5 − 0.866i)19-s + (−0.866 + 0.5i)22-s − i·23-s + (0.866 − 0.5i)28-s + (−0.5 + 0.866i)29-s + (−0.5 + 0.866i)31-s + (0.866 − 0.5i)32-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)2-s + (0.5 + 0.866i)4-s − i·7-s − i·8-s + (0.5 − 0.866i)11-s + (−0.5 + 0.866i)14-s + (−0.5 + 0.866i)16-s + (−0.866 − 0.5i)17-s + (0.5 − 0.866i)19-s + (−0.866 + 0.5i)22-s − i·23-s + (0.866 − 0.5i)28-s + (−0.5 + 0.866i)29-s + (−0.5 + 0.866i)31-s + (0.866 − 0.5i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.839 - 0.543i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.839 - 0.543i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $-0.839 - 0.543i$
Analytic conductor: \(2.71672\)
Root analytic conductor: \(2.71672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{585} (308, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 585,\ (0:\ ),\ -0.839 - 0.543i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1857011711 - 0.6286254624i\)
\(L(\frac12)\) \(\approx\) \(0.1857011711 - 0.6286254624i\)
\(L(1)\) \(\approx\) \(0.5875140663 - 0.3193041292i\)
\(L(1)\) \(\approx\) \(0.5875140663 - 0.3193041292i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.866 - 0.5i)T \)
7 \( 1 - iT \)
11 \( 1 + (0.5 - 0.866i)T \)
17 \( 1 + (-0.866 - 0.5i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 - iT \)
29 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (-0.866 + 0.5i)T \)
41 \( 1 - T \)
43 \( 1 + iT \)
47 \( 1 + (0.866 - 0.5i)T \)
53 \( 1 - iT \)
59 \( 1 + (-0.5 - 0.866i)T \)
61 \( 1 + T \)
67 \( 1 - iT \)
71 \( 1 + (0.5 - 0.866i)T \)
73 \( 1 + iT \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (-0.866 + 0.5i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 - iT \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−23.74605846951015691029777594612, −22.71635065837080906033398745429, −21.97550424533848704545040745260, −20.78493595526132542855877263530, −20.07627619183779143766092653414, −19.127891224804513928727500612442, −18.52248584902685087871428174819, −17.586200127304920825376715251524, −17.056195562829606339354593346328, −15.86729667197849794361130902844, −15.284228225637305160371322685315, −14.62041867865446500488610434340, −13.4828236698302525996289472706, −12.20275803489736661762762626538, −11.56759914384857753390327702305, −10.45583012347000451320055688569, −9.49896748899254330458754203738, −8.94327208449219045311756853222, −7.91066885163775271903700913283, −7.05664909575844959838771958859, −6.01527148744935249203598073973, −5.32254010375498662491010240240, −3.948550103630674774211924339, −2.3423242476151884869143047958, −1.5830938252538300121661025212, 0.44973101952388478486191751611, 1.58759361979820211405747286730, 2.94892892022770070911236038046, 3.77958309521943621199384703305, 4.94644279266320111967331569481, 6.64271551560317559655571796628, 7.063458368742508769803086790144, 8.30587081691572787791611059884, 8.991165096965095633960967881523, 9.95820702529529514564662235174, 10.93140566116746578929278050253, 11.35020724319537097706564240693, 12.52441015513644588117838594418, 13.44547018436196714757658752995, 14.216294006827666479389616811747, 15.5839435436453392819351215672, 16.41718082799285615260232777902, 17.015530772403838297464577344382, 17.91080206524369285809112675543, 18.646364278339387254092805697204, 19.722345976973930047204605473610, 20.08483480543146511861746130632, 20.95967366008237466222216586066, 21.973577891603136245900281038704, 22.53811841803544769897018840210

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