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Properties

Label 2-285-285.254-c0-0-0
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

\(L(\frac{1}{2})\) \(\approx\) \(0.7321303521\)
\(L(\frac12)\) \(\approx\) \(0.7321303521\)
\(L(1)\) 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

−11.51751798341448375312340509427, −10.78843812725046325641895246764, −10.36408278704960276756772240631, −9.267718195822286294070247811545, −8.837912320069298530516158571661, −7.19728086862499043503482091283, −6.04440580782775814248544078125, −4.66448061686693035431336731252, −3.06961682813150579376391297873, −2.34662064095576187253817136458, 1.83148390906113682803087360546, 3.56272600454641070471240978617, 5.48734179696336457733436721601, 6.35439706637639533905460966469, 7.36588345451381232017501391620, 8.241891285257967210315647375124, 8.895193985619595430857866418756, 9.696230119481089221587304749717, 11.31467196330016449068808881213, 12.41610330087942861181396474560

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