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Properties

Label 2-585-117.103-c1-0-30
Degree $2$
Conductor $585$
Sign $0.882 - 0.470i$
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  + (1.16 − 0.673i)2-s + (0.619 + 1.61i)3-s + (−0.0923 + 0.159i)4-s + (−0.866 − 0.5i)5-s + (1.81 + 1.46i)6-s + (4.07 − 2.35i)7-s + 2.94i·8-s + (−2.23 + 2.00i)9-s − 1.34·10-s + (1.11 − 0.641i)11-s + (−0.315 − 0.0501i)12-s + (2.91 − 2.12i)13-s + (3.16 − 5.48i)14-s + (0.271 − 1.71i)15-s + (1.79 + 3.11i)16-s − 0.689·17-s + ⋯
L(s)  = 1  + (0.825 − 0.476i)2-s + (0.357 + 0.933i)3-s + (−0.0461 + 0.0799i)4-s + (−0.387 − 0.223i)5-s + (0.740 + 0.599i)6-s + (1.53 − 0.888i)7-s + 1.04i·8-s + (−0.743 + 0.668i)9-s − 0.426·10-s + (0.334 − 0.193i)11-s + (−0.0912 − 0.0144i)12-s + (0.808 − 0.589i)13-s + (0.846 − 1.46i)14-s + (0.0701 − 0.441i)15-s + (0.449 + 0.778i)16-s − 0.167·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.882 - 0.470i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.882 - 0.470i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $0.882 - 0.470i$
Analytic conductor: \(4.67124\)
Root analytic conductor: \(2.16130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{585} (571, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 585,\ (\ :1/2),\ 0.882 - 0.470i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.49081 + 0.621830i\)
\(L(\frac12)\) \(\approx\) \(2.49081 + 0.621830i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.619 - 1.61i)T \)
5 \( 1 + (0.866 + 0.5i)T \)
13 \( 1 + (-2.91 + 2.12i)T \)
good2 \( 1 + (-1.16 + 0.673i)T + (1 - 1.73i)T^{2} \)
7 \( 1 + (-4.07 + 2.35i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.11 + 0.641i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + 0.689T + 17T^{2} \)
19 \( 1 - 6.46iT - 19T^{2} \)
23 \( 1 + (2.10 - 3.64i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.78 - 4.82i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.646 - 0.373i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 8.06iT - 37T^{2} \)
41 \( 1 + (6.39 + 3.69i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.42 + 9.39i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-6.50 + 3.75i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 5.44T + 53T^{2} \)
59 \( 1 + (0.357 + 0.206i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.91 + 5.05i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-9.45 - 5.46i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 5.80iT - 71T^{2} \)
73 \( 1 + 9.51iT - 73T^{2} \)
79 \( 1 + (2.41 + 4.19i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.46 + 1.42i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 10.7iT - 89T^{2} \)
97 \( 1 + (-1.55 + 0.895i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.85002314800411657829680167833, −10.30236051227865850077239920071, −8.770475278803200368699507076260, −8.275985037963830932947598059225, −7.54197095103210719280311064391, −5.61230311728437338524041518662, −4.92570596114730018961688373869, −3.88133952078633418392490657716, −3.59548419177811352878046240053, −1.78711356302551042883102475481, 1.35679341641257511421512845191, 2.71801540051253919762292980061, 4.26060150519678384192015372872, 5.02670067313606327519823361476, 6.24706348744326186984073195041, 6.75903447169759620446665698519, 7.976377774977293099238453951014, 8.570100893041482839052220485276, 9.505117732295602479908620904462, 11.05357347548656250059118712183

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