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Properties

Label 2-285-19.7-c1-0-1
Degree $2$
Conductor $285$
Sign $-0.990 - 0.138i$
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  + (1.34 + 2.33i)2-s + (0.5 + 0.866i)3-s + (−2.62 + 4.54i)4-s + (−0.5 − 0.866i)5-s + (−1.34 + 2.33i)6-s − 0.797·7-s − 8.73·8-s + (−0.499 + 0.866i)9-s + (1.34 − 2.33i)10-s + 2.59·11-s − 5.24·12-s + (1.39 − 2.42i)13-s + (−1.07 − 1.85i)14-s + (0.499 − 0.866i)15-s + (−6.50 − 11.2i)16-s + (2.88 + 4.99i)17-s + ⋯
L(s)  = 1  + (0.951 + 1.64i)2-s + (0.288 + 0.499i)3-s + (−1.31 + 2.27i)4-s + (−0.223 − 0.387i)5-s + (−0.549 + 0.951i)6-s − 0.301·7-s − 3.08·8-s + (−0.166 + 0.288i)9-s + (0.425 − 0.737i)10-s + 0.781·11-s − 1.51·12-s + (0.387 − 0.671i)13-s + (−0.286 − 0.496i)14-s + (0.129 − 0.223i)15-s + (−1.62 − 2.81i)16-s + (0.700 + 1.21i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 - 0.138i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 - 0.138i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(285\)    =    \(3 \cdot 5 \cdot 19\)
Sign: $-0.990 - 0.138i$
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: \((2,\ 285,\ (\ :1/2),\ -0.990 - 0.138i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.134497 + 1.93034i\)
\(L(\frac12)\) \(\approx\) \(0.134497 + 1.93034i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.45 - 3.60i)T \)
good2 \( 1 + (-1.34 - 2.33i)T + (-1 + 1.73i)T^{2} \)
7 \( 1 + 0.797T + 7T^{2} \)
11 \( 1 - 2.59T + 11T^{2} \)
13 \( 1 + (-1.39 + 2.42i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.88 - 4.99i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-1.55 + 2.68i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.39 + 4.14i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 9.48T + 31T^{2} \)
37 \( 1 - 7.69T + 37T^{2} \)
41 \( 1 + (3.69 + 6.39i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.39 - 2.42i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5.53 + 9.59i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.43 + 7.68i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.540 + 0.935i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.03 - 3.52i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.88 - 11.9i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-5.98 - 10.3i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (4.25 + 7.36i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.24 + 5.61i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 2.79T + 83T^{2} \)
89 \( 1 + (-6.67 + 11.5i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1 + 1.73i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.69415200299867101792125005337, −11.75970025691154879328978152878, −10.15654784872693856444817381630, −8.947699807452508867294113079090, −8.256676196368136119079200535230, −7.36680119880534778790362773524, −6.12029108481249276253445540752, −5.42261706731170238731181467474, −4.10886318487099630596364453985, −3.47550689023340287357751285575, 1.25537222928099087428126062685, 2.78325345107675534957266359408, 3.61162889077887765419403512765, 4.86246996830041606712034420148, 6.14258442082496962963217172714, 7.30027938481784856662981426132, 9.163157992671167921708902179290, 9.516399748202114159008220827529, 10.91786474954719103009895158835, 11.50458905630007893045012711791

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