Properties

Label 2-585-9.4-c1-0-12
Degree $2$
Conductor $585$
Sign $-0.458 - 0.888i$
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  + (1.26 + 2.18i)2-s + (−0.259 − 1.71i)3-s + (−2.18 + 3.79i)4-s + (0.5 − 0.866i)5-s + (3.41 − 2.73i)6-s + (0.799 + 1.38i)7-s − 6.00·8-s + (−2.86 + 0.890i)9-s + 2.52·10-s + (2.68 + 4.65i)11-s + (7.05 + 2.76i)12-s + (−0.5 + 0.866i)13-s + (−2.01 + 3.49i)14-s + (−1.61 − 0.631i)15-s + (−3.20 − 5.54i)16-s + 3.08·17-s + ⋯
L(s)  = 1  + (0.892 + 1.54i)2-s + (−0.150 − 0.988i)3-s + (−1.09 + 1.89i)4-s + (0.223 − 0.387i)5-s + (1.39 − 1.11i)6-s + (0.302 + 0.523i)7-s − 2.12·8-s + (−0.954 + 0.296i)9-s + 0.798·10-s + (0.811 + 1.40i)11-s + (2.03 + 0.797i)12-s + (−0.138 + 0.240i)13-s + (−0.539 + 0.934i)14-s + (−0.416 − 0.162i)15-s + (−0.800 − 1.38i)16-s + 0.748·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.458 - 0.888i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.458 - 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $-0.458 - 0.888i$
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: \((2,\ 585,\ (\ :1/2),\ -0.458 - 0.888i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.13082 + 1.85473i\)
\(L(\frac12)\) \(\approx\) \(1.13082 + 1.85473i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.259 + 1.71i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (-1.26 - 2.18i)T + (-1 + 1.73i)T^{2} \)
7 \( 1 + (-0.799 - 1.38i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.68 - 4.65i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 - 3.08T + 17T^{2} \)
19 \( 1 - 3.72T + 19T^{2} \)
23 \( 1 + (0.847 - 1.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.320 + 0.555i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.79 - 4.83i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 0.445T + 37T^{2} \)
41 \( 1 + (-3.19 + 5.53i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.82 + 4.90i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.70 + 9.88i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 10.3T + 53T^{2} \)
59 \( 1 + (-4.67 + 8.09i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.23 - 5.60i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.64 + 13.2i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 5.56T + 71T^{2} \)
73 \( 1 - 11.7T + 73T^{2} \)
79 \( 1 + (-2.66 - 4.60i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.87 - 8.44i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 15.7T + 89T^{2} \)
97 \( 1 + (-0.456 - 0.791i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.52512181344680607781583307041, −9.741951484543492486516956276655, −8.809069682007307722762905470485, −7.959226295612815494859975187762, −7.15972784881198810344038827704, −6.58815510700008228266917398395, −5.46418578504536311771041163435, −5.01478320726098214840256094356, −3.63648251777088077974330507825, −1.85723959657610170957568931564, 1.05779620647284117390699219421, 2.88799888915988077487202246710, 3.54768059190340504073813725205, 4.43015198146304243477972746161, 5.47215282547699195542044719296, 6.19869986091414763881249546215, 7.997004219965508362648807197350, 9.303785997791341035856367731124, 9.824336403893517542425290269867, 10.70942205295751003626452418417

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