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Properties

Label 2-285-19.11-c1-0-6
Degree $2$
Conductor $285$
Sign $0.910 + 0.412i$
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  + (−1.20 + 2.09i)2-s + (0.5 − 0.866i)3-s + (−1.91 − 3.31i)4-s + (0.5 − 0.866i)5-s + (1.20 + 2.09i)6-s − 3.82·7-s + 4.41·8-s + (−0.499 − 0.866i)9-s + (1.20 + 2.09i)10-s + 2.82·11-s − 3.82·12-s + (−1.91 − 3.31i)13-s + (4.62 − 8.00i)14-s + (−0.499 − 0.866i)15-s + (−1.49 + 2.59i)16-s + (3.41 − 5.91i)17-s + ⋯
L(s)  = 1  + (−0.853 + 1.47i)2-s + (0.288 − 0.499i)3-s + (−0.957 − 1.65i)4-s + (0.223 − 0.387i)5-s + (0.492 + 0.853i)6-s − 1.44·7-s + 1.56·8-s + (−0.166 − 0.288i)9-s + (0.381 + 0.661i)10-s + 0.852·11-s − 1.10·12-s + (−0.530 − 0.919i)13-s + (1.23 − 2.13i)14-s + (−0.129 − 0.223i)15-s + (−0.374 + 0.649i)16-s + (0.828 − 1.43i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 + 0.412i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.910 + 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(285\)    =    \(3 \cdot 5 \cdot 19\)
Sign: $0.910 + 0.412i$
Analytic conductor: \(2.27573\)
Root analytic conductor: \(1.50855\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{285} (106, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 285,\ (\ :1/2),\ 0.910 + 0.412i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.627494 - 0.135618i\)
\(L(\frac12)\) \(\approx\) \(0.627494 - 0.135618i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4 + 1.73i)T \)
good2 \( 1 + (1.20 - 2.09i)T + (-1 - 1.73i)T^{2} \)
7 \( 1 + 3.82T + 7T^{2} \)
11 \( 1 - 2.82T + 11T^{2} \)
13 \( 1 + (1.91 + 3.31i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.41 + 5.91i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (2.41 + 4.18i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.828 - 1.43i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 5T + 31T^{2} \)
37 \( 1 + 7.82T + 37T^{2} \)
41 \( 1 + (-1.41 + 2.44i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.08 - 1.88i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.41 - 7.64i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1 - 1.73i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.15 + 12.3i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.74 - 9.94i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (5 - 8.66i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-3.74 + 6.48i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-7.32 + 12.6i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 8T + 83T^{2} \)
89 \( 1 + (-2.24 - 3.88i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3 + 5.19i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.02883543127001932954314830310, −10.19543432613413662953580693655, −9.447689955570824781350772477656, −8.943927970887545696959726527177, −7.67873442247091151783772898953, −7.02607939682509670102950005724, −6.13782312693259007017630299683, −5.18213437904242541020277990687, −3.13314731433638830095454856736, −0.62688325020116033969017506592, 1.83446124223657995218546554359, 3.33620189312162287666657318407, 3.83022655100526189477527971678, 5.93176973212319772889184512126, 7.25369428120164197946998489691, 8.624376751737401864417724159078, 9.554119066052589656208921892721, 9.854015220012553664062158910214, 10.69016263288119220354999580673, 11.85284537712547790846711146360

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