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Properties

Label 2-285-285.254-c0-0-1
Degree $2$
Conductor $285$
Sign $0.977 + 0.211i$
Analytic cond. $0.142233$
Root an. cond. $0.377138$
Motivic weight $0$
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 + (−0.5 − 0.866i)3-s + (−0.5 − 0.866i)5-s + (0.499 − 0.866i)6-s + 8-s + (−0.499 + 0.866i)9-s + (0.499 − 0.866i)10-s + (−0.499 + 0.866i)15-s + (0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s − 0.999·18-s + (−0.5 − 0.866i)19-s + (−1 + 1.73i)23-s + (−0.499 − 0.866i)24-s + (−0.499 + 0.866i)25-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.5 − 0.866i)5-s + (0.499 − 0.866i)6-s + 8-s + (−0.499 + 0.866i)9-s + (0.499 − 0.866i)10-s + (−0.499 + 0.866i)15-s + (0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s − 0.999·18-s + (−0.5 − 0.866i)19-s + (−1 + 1.73i)23-s + (−0.499 − 0.866i)24-s + (−0.499 + 0.866i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 + 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 + 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(285\)    =    \(3 \cdot 5 \cdot 19\)
Sign: $0.977 + 0.211i$
Analytic conductor: \(0.142233\)
Root analytic conductor: \(0.377138\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{285} (254, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 285,\ (\ :0),\ 0.977 + 0.211i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8422124455\)
\(L(\frac12)\) \(\approx\) \(0.8422124455\)
\(L(1)\) 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 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + T + T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + T + T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (0.5 - 0.866i)T^{2} \)
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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

−12.15785540250567019700707276400, −11.35666718199227960122581464651, −10.30689499721270675896048712654, −8.803629458937160843920717360513, −7.78163581662184131164455461850, −7.15776271379587431554235447013, −5.91529119249644692577341081957, −5.31867474827806099242279641968, −4.06834559661622766405987775895, −1.64188115149340304564961950889, 2.61832966739499539172622774860, 3.72184790387501559781620520725, 4.50532264963297987329073627295, 5.93179847792573482315693980928, 7.14408847947876475029191356184, 8.280811080948241475280318091472, 9.760318752491839606623725071076, 10.58440104013704425969792959866, 11.06411726449873000039909340373, 12.07697521202778649935892106894

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