Properties

Label 2-285-285.119-c0-0-1
Degree $2$
Conductor $285$
Sign $0.990 + 0.135i$
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  + (1.43 − 0.524i)2-s + (−0.173 + 0.984i)3-s + (1.03 − 0.866i)4-s + (−0.766 − 0.642i)5-s + (0.266 + 1.50i)6-s + (0.266 − 0.460i)8-s + (−0.939 − 0.342i)9-s + (−1.43 − 0.524i)10-s + (0.673 + 1.16i)12-s + (0.766 − 0.642i)15-s + (−0.0923 + 0.524i)16-s + (−1.76 + 0.642i)17-s − 1.53·18-s + (0.173 − 0.984i)19-s − 1.34·20-s + ⋯
L(s)  = 1  + (1.43 − 0.524i)2-s + (−0.173 + 0.984i)3-s + (1.03 − 0.866i)4-s + (−0.766 − 0.642i)5-s + (0.266 + 1.50i)6-s + (0.266 − 0.460i)8-s + (−0.939 − 0.342i)9-s + (−1.43 − 0.524i)10-s + (0.673 + 1.16i)12-s + (0.766 − 0.642i)15-s + (−0.0923 + 0.524i)16-s + (−1.76 + 0.642i)17-s − 1.53·18-s + (0.173 − 0.984i)19-s − 1.34·20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.135i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.135i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(285\)    =    \(3 \cdot 5 \cdot 19\)
Sign: $0.990 + 0.135i$
Analytic conductor: \(0.142233\)
Root analytic conductor: \(0.377138\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{285} (119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 285,\ (\ :0),\ 0.990 + 0.135i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.254450995\)
\(L(\frac12)\) \(\approx\) \(1.254450995\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.173 - 0.984i)T \)
5 \( 1 + (0.766 + 0.642i)T \)
19 \( 1 + (-0.173 + 0.984i)T \)
good2 \( 1 + (-1.43 + 0.524i)T + (0.766 - 0.642i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.939 - 0.342i)T^{2} \)
17 \( 1 + (1.76 - 0.642i)T + (0.766 - 0.642i)T^{2} \)
23 \( 1 + (-0.766 + 0.642i)T + (0.173 - 0.984i)T^{2} \)
29 \( 1 + (-0.766 - 0.642i)T^{2} \)
31 \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (0.939 + 0.342i)T^{2} \)
43 \( 1 + (-0.173 - 0.984i)T^{2} \)
47 \( 1 + (-1.43 - 0.524i)T + (0.766 + 0.642i)T^{2} \)
53 \( 1 + (-1.43 + 1.20i)T + (0.173 - 0.984i)T^{2} \)
59 \( 1 + (-0.766 + 0.642i)T^{2} \)
61 \( 1 + (0.766 - 0.642i)T + (0.173 - 0.984i)T^{2} \)
67 \( 1 + (-0.766 - 0.642i)T^{2} \)
71 \( 1 + (-0.173 - 0.984i)T^{2} \)
73 \( 1 + (0.939 + 0.342i)T^{2} \)
79 \( 1 + (0.173 - 0.984i)T + (-0.939 - 0.342i)T^{2} \)
83 \( 1 + (-0.173 - 0.300i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 + (0.939 - 0.342i)T^{2} \)
97 \( 1 + (-0.766 + 0.642i)T^{2} \)
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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

−12.04974984040644359312944856215, −11.17055883961491829220116965573, −10.77331013510667715622969303243, −9.189338096976710823030278747713, −8.485520438667346134609467973120, −6.74414085738013175794918733852, −5.49348741541786826041408957845, −4.54861483542438634931694193169, −4.04411960311907984571054516076, −2.71484977217962682589049111425, 2.60031055058779045732555514958, 3.81251638633129187039886799262, 5.06412496625005352757204882521, 6.20547253459175653183572755737, 6.99288876793721282594679364652, 7.62084970447560224560039398688, 8.921763971370198173318857569383, 10.71387020893963286388797957291, 11.63055144168683879737137454858, 12.17056755729535196012655792889

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