Properties

Label 2-585-9.4-c1-0-39
Degree $2$
Conductor $585$
Sign $0.173 + 0.984i$
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  + (0.5 + 0.866i)2-s − 1.73i·3-s + (0.500 − 0.866i)4-s + (0.5 − 0.866i)5-s + (1.49 − 0.866i)6-s + (−1 − 1.73i)7-s + 3·8-s − 2.99·9-s + 0.999·10-s + (0.5 + 0.866i)11-s + (−1.49 − 0.866i)12-s + (0.5 − 0.866i)13-s + (0.999 − 1.73i)14-s + (−1.49 − 0.866i)15-s + (0.500 + 0.866i)16-s + 2·17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s − 0.999i·3-s + (0.250 − 0.433i)4-s + (0.223 − 0.387i)5-s + (0.612 − 0.353i)6-s + (−0.377 − 0.654i)7-s + 1.06·8-s − 0.999·9-s + 0.316·10-s + (0.150 + 0.261i)11-s + (−0.433 − 0.250i)12-s + (0.138 − 0.240i)13-s + (0.267 − 0.462i)14-s + (−0.387 − 0.223i)15-s + (0.125 + 0.216i)16-s + 0.485·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $0.173 + 0.984i$
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: \((2,\ 585,\ (\ :1/2),\ 0.173 + 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.42564 - 1.19626i\)
\(L(\frac12)\) \(\approx\) \(1.42564 - 1.19626i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 1.73iT \)
5 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (-0.5 - 0.866i)T + (-1 + 1.73i)T^{2} \)
7 \( 1 + (1 + 1.73i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 - 2T + 17T^{2} \)
19 \( 1 + 3T + 19T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.5 + 4.33i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 5T + 37T^{2} \)
41 \( 1 + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4 - 6.92i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1 - 1.73i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 14T + 53T^{2} \)
59 \( 1 + (-4.5 + 7.79i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (7 - 12.1i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 6T + 73T^{2} \)
79 \( 1 + (-6 - 10.3i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5 - 8.66i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 2T + 89T^{2} \)
97 \( 1 + (0.5 + 0.866i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.51811195697517297734044584970, −9.677481595227111620669874779139, −8.460876341539892122819138262147, −7.56466385163742694478419648572, −6.84334040578923709037252472116, −6.06711973469745507013148682859, −5.26612833496443986194770641292, −3.97057079452193291999780406948, −2.26531065152294420070403497531, −0.969386349372659091894649488479, 2.21732948289237067147445627025, 3.22852679528122311446459928508, 3.98285346276636788456939161534, 5.19545965994405052571672526068, 6.17574306498307210053639030406, 7.30949724160604065071813470644, 8.556570091677377216414156840985, 9.204035616814526049861858735042, 10.39112812815007298568815751325, 10.73630668251011404097820302396

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