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Properties

Label 2-285-19.7-c1-0-2
Degree $2$
Conductor $285$
Sign $-0.658 - 0.752i$
Analytic cond. $2.27573$
Root an. cond. $1.50855$
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 + (0.5 + 0.866i)3-s + (−0.999 + 1.73i)4-s + (0.5 + 0.866i)5-s + (−0.999 + 1.73i)6-s − 2·7-s + (−0.499 + 0.866i)9-s + (−0.999 + 1.73i)10-s + 11-s − 1.99·12-s + (−1 + 1.73i)13-s + (−2 − 3.46i)14-s + (−0.499 + 0.866i)15-s + (1.99 + 3.46i)16-s + (−1 − 1.73i)17-s − 1.99·18-s + ⋯
L(s)  = 1  + (0.707 + 1.22i)2-s + (0.288 + 0.499i)3-s + (−0.499 + 0.866i)4-s + (0.223 + 0.387i)5-s + (−0.408 + 0.707i)6-s − 0.755·7-s + (−0.166 + 0.288i)9-s + (−0.316 + 0.547i)10-s + 0.301·11-s − 0.577·12-s + (−0.277 + 0.480i)13-s + (−0.534 − 0.925i)14-s + (−0.129 + 0.223i)15-s + (0.499 + 0.866i)16-s + (−0.242 − 0.420i)17-s − 0.471·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.658 - 0.752i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.658 - 0.752i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(285\)    =    \(3 \cdot 5 \cdot 19\)
Sign: $-0.658 - 0.752i$
Analytic conductor: \(2.27573\)
Root analytic conductor: \(1.50855\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{285} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 285,\ (\ :1/2),\ -0.658 - 0.752i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.808678 + 1.78345i\)
\(L(\frac12)\) \(\approx\) \(0.808678 + 1.78345i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (-0.5 + 4.33i)T \)
good2 \( 1 + (-1 - 1.73i)T + (-1 + 1.73i)T^{2} \)
7 \( 1 + 2T + 7T^{2} \)
11 \( 1 - T + 11T^{2} \)
13 \( 1 + (1 - 1.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (1 + 1.73i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-2 + 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.5 + 4.33i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 9T + 31T^{2} \)
37 \( 1 + 6T + 37T^{2} \)
41 \( 1 + (3 + 5.19i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5 - 8.66i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1 + 1.73i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.5 + 6.06i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.5 + 6.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4 - 6.92i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.5 + 2.59i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-1 - 1.73i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.5 - 9.52i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 6T + 83T^{2} \)
89 \( 1 + (7.5 - 12.9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (4 + 6.92i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.46209941022812208600273517696, −11.21198045734166541035629071582, −10.12559445741482174879000747059, −9.249830612000781248658361048083, −8.149573713843049867930795382755, −6.87269569677938404522904086291, −6.44140976779586295962524123115, −5.11183917912864208719460375644, −4.19230080735664690705498155337, −2.79162727271473692132161533456, 1.39979474596827242053889525407, 2.81806558401859571988235238245, 3.79557217055336642956445750817, 5.13903439692708512630776699117, 6.35392494596635930365160337874, 7.62215301032654969398137388535, 8.823268492205497817692326779418, 9.910908038325248261984370336318, 10.59004637505799710566969515085, 11.93696717434466127438971427028

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