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Properties

Label 2-585-13.3-c1-0-18
Degree $2$
Conductor $585$
Sign $0.964 - 0.265i$
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  + (0.651 + 1.12i)2-s + (0.151 − 0.262i)4-s + 5-s + (0.5 − 0.866i)7-s + 3·8-s + (0.651 + 1.12i)10-s + (−2.80 − 4.85i)11-s + 3.60·13-s + 1.30·14-s + (1.65 + 2.86i)16-s + (0.197 − 0.341i)17-s + (−0.802 + 1.39i)19-s + (0.151 − 0.262i)20-s + (3.65 − 6.32i)22-s + (−1.5 − 2.59i)23-s + ⋯
L(s)  = 1  + (0.460 + 0.797i)2-s + (0.0756 − 0.131i)4-s + 0.447·5-s + (0.188 − 0.327i)7-s + 1.06·8-s + (0.205 + 0.356i)10-s + (−0.845 − 1.46i)11-s + 1.00·13-s + 0.348·14-s + (0.412 + 0.715i)16-s + (0.0478 − 0.0828i)17-s + (−0.184 + 0.318i)19-s + (0.0338 − 0.0586i)20-s + (0.778 − 1.34i)22-s + (−0.312 − 0.541i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 - 0.265i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.964 - 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $0.964 - 0.265i$
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: \((2,\ 585,\ (\ :1/2),\ 0.964 - 0.265i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.22563 + 0.300272i\)
\(L(\frac12)\) \(\approx\) \(2.22563 + 0.300272i\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.60T \)
good2 \( 1 + (-0.651 - 1.12i)T + (-1 + 1.73i)T^{2} \)
7 \( 1 + (-0.5 + 0.866i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.80 + 4.85i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-0.197 + 0.341i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.802 - 1.39i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.5 + 2.59i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.10 - 7.11i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + (-1.80 - 3.12i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.5 - 2.59i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.10 - 3.64i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 5.21T + 47T^{2} \)
53 \( 1 + 11.2T + 53T^{2} \)
59 \( 1 + (-5.40 + 9.36i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.5 - 6.06i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (8.40 - 14.5i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 15.2T + 73T^{2} \)
79 \( 1 + 9.21T + 79T^{2} \)
83 \( 1 + 5.21T + 83T^{2} \)
89 \( 1 + (-4.10 - 7.11i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-7.80 + 13.5i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.78723673344519780126440776093, −10.05322623360650429975988043255, −8.672903650884389366505312317355, −8.053640255039971521429320980125, −6.94035521265730718962614189013, −6.04239136989103537315520057443, −5.50098449856334993908599171837, −4.37498755968060647550406637600, −3.03076952136180675816034940387, −1.30514420746995602798904358704, 1.77661219824126777575458276923, 2.61519630802696832619676073954, 3.93897582977018517080391798143, 4.85654305130053525472655843335, 5.92618970641545711914155846601, 7.18248499983267701018513766325, 7.941621394492587946477858563754, 9.057906034885089972828216129413, 10.12237141940256213321673252880, 10.66324325828263939207709707771

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