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Properties

Label 2-285-19.7-c1-0-6
Degree $2$
Conductor $285$
Sign $0.167 - 0.985i$
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  + (0.823 + 1.42i)2-s + (0.5 + 0.866i)3-s + (−0.355 + 0.616i)4-s + (−0.5 − 0.866i)5-s + (−0.823 + 1.42i)6-s + 4.47·7-s + 2.12·8-s + (−0.499 + 0.866i)9-s + (0.823 − 1.42i)10-s − 3.44·11-s − 0.711·12-s + (−1.23 + 2.14i)13-s + (3.68 + 6.38i)14-s + (0.499 − 0.866i)15-s + (2.45 + 4.25i)16-s + (−3.81 − 6.60i)17-s + ⋯
L(s)  = 1  + (0.582 + 1.00i)2-s + (0.288 + 0.499i)3-s + (−0.177 + 0.308i)4-s + (−0.223 − 0.387i)5-s + (−0.336 + 0.582i)6-s + 1.69·7-s + 0.750·8-s + (−0.166 + 0.288i)9-s + (0.260 − 0.450i)10-s − 1.03·11-s − 0.205·12-s + (−0.343 + 0.595i)13-s + (0.985 + 1.70i)14-s + (0.129 − 0.223i)15-s + (0.614 + 1.06i)16-s + (−0.925 − 1.60i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.167 - 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.167 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(285\)    =    \(3 \cdot 5 \cdot 19\)
Sign: $0.167 - 0.985i$
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: \((2,\ 285,\ (\ :1/2),\ 0.167 - 0.985i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.56662 + 1.32225i\)
\(L(\frac12)\) \(\approx\) \(1.56662 + 1.32225i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.67 - 2.34i)T \)
good2 \( 1 + (-0.823 - 1.42i)T + (-1 + 1.73i)T^{2} \)
7 \( 1 - 4.47T + 7T^{2} \)
11 \( 1 + 3.44T + 11T^{2} \)
13 \( 1 + (1.23 - 2.14i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (3.81 + 6.60i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (1.93 - 3.35i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.36 + 7.56i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 0.422T + 31T^{2} \)
37 \( 1 - 3.90T + 37T^{2} \)
41 \( 1 + (2.64 + 4.58i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.23 + 2.14i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.338 + 0.586i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.74 - 9.95i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.26 - 7.38i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.10 - 7.10i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.81 + 8.34i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.92 - 3.33i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (8.39 + 14.5i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.06 + 10.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 2.03T + 83T^{2} \)
89 \( 1 + (-1.57 + 2.72i)T + (-44.5 - 77.0i)T^{2} \)
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

−11.96197388617701526746606014645, −11.15040424966026167953098096241, −10.22038720886826779559021306049, −8.873140809791690877655481242755, −7.940500840805366092478421042990, −7.35587583783416384181053622834, −5.81782966204360428695486092271, −4.74350920747302401701508932932, −4.45379776604968537275642797750, −2.14996073875566403962793464160, 1.78824925461140334434166787551, 2.73365662766199425364232749833, 4.22529560552548679887262817609, 5.12785441925422112169156265582, 6.78196149569131953577440051111, 8.106327373668018987600072741612, 8.280343834159096355480794343882, 10.33934182207109868267802008707, 10.88912206209433786795643896734, 11.53474361046311750747017867908

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