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Properties

Label 1-585-585.302-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: $\chi_{585} (302, \cdot )$
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

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

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