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Properties

Label 2-585-13.3-c1-0-8
Degree $2$
Conductor $585$
Sign $-0.859 - 0.511i$
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 + 1.73i)2-s + (−0.999 + 1.73i)4-s + 5-s + (−2.5 + 4.33i)7-s + (1 + 1.73i)10-s + (1 + 1.73i)11-s + (−2.5 − 2.59i)13-s − 10·14-s + (1.99 + 3.46i)16-s + (1 − 1.73i)17-s + (−0.999 + 1.73i)20-s + (−1.99 + 3.46i)22-s + (3 + 5.19i)23-s + 25-s + (2 − 6.92i)26-s + ⋯
L(s)  = 1  + (0.707 + 1.22i)2-s + (−0.499 + 0.866i)4-s + 0.447·5-s + (−0.944 + 1.63i)7-s + (0.316 + 0.547i)10-s + (0.301 + 0.522i)11-s + (−0.693 − 0.720i)13-s − 2.67·14-s + (0.499 + 0.866i)16-s + (0.242 − 0.420i)17-s + (−0.223 + 0.387i)20-s + (−0.426 + 0.738i)22-s + (0.625 + 1.08i)23-s + 0.200·25-s + (0.392 − 1.35i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.859 - 0.511i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.859 - 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $-0.859 - 0.511i$
Analytic conductor: \(4.67124\)
Root analytic conductor: \(2.16130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{585} (406, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 585,\ (\ :1/2),\ -0.859 - 0.511i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.537942 + 1.95734i\)
\(L(\frac12)\) \(\approx\) \(0.537942 + 1.95734i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 - T \)
13 \( 1 + (2.5 + 2.59i)T \)
good2 \( 1 + (-1 - 1.73i)T + (-1 + 1.73i)T^{2} \)
7 \( 1 + (2.5 - 4.33i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1 + 1.73i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3 - 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2 + 3.46i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 7T + 31T^{2} \)
37 \( 1 + (-1 - 1.73i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3 - 5.19i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 8T + 47T^{2} \)
53 \( 1 - 4T + 53T^{2} \)
59 \( 1 + (-6 + 10.3i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.5 + 11.2i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.5 - 6.06i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-6 + 10.3i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 15T + 73T^{2} \)
79 \( 1 - 3T + 79T^{2} \)
83 \( 1 + 8T + 83T^{2} \)
89 \( 1 + (-7 - 12.1i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.5 + 4.33i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.18980137620173473128583712908, −9.744780878214181370781678972733, −9.388679003272540244048190508296, −8.210207343397531711686023667978, −7.22891461842650357977994977254, −6.40511687785417313766058974595, −5.53210491696686668663124785236, −5.12588428946771664653742931642, −3.53026721314585480121250632595, −2.27699844307086359268012777095, 0.949416655390928422297577573260, 2.43863148074310108346230017260, 3.65629075785898758819036233409, 4.19776069111908129606904026019, 5.45958830313248885414808741965, 6.76994410642132475742346448226, 7.38678411475945676825566959440, 8.983216747599238767323641699146, 9.841940536376980381046873061917, 10.58466704533406378544303841166

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