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Properties

Label 2-285-19.11-c1-0-1
Degree $2$
Conductor $285$
Sign $-0.0977 - 0.995i$
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  + (−0.5 + 0.866i)3-s + (1 + 1.73i)4-s + (−0.5 + 0.866i)5-s + 2·7-s + (−0.499 − 0.866i)9-s − 3·11-s − 1.99·12-s + (2 + 3.46i)13-s + (−0.499 − 0.866i)15-s + (−1.99 + 3.46i)16-s + (−3.5 − 2.59i)19-s − 1.99·20-s + (−1 + 1.73i)21-s + (3 + 5.19i)23-s + (−0.499 − 0.866i)25-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + (0.5 + 0.866i)4-s + (−0.223 + 0.387i)5-s + 0.755·7-s + (−0.166 − 0.288i)9-s − 0.904·11-s − 0.577·12-s + (0.554 + 0.960i)13-s + (−0.129 − 0.223i)15-s + (−0.499 + 0.866i)16-s + (−0.802 − 0.596i)19-s − 0.447·20-s + (−0.218 + 0.377i)21-s + (0.625 + 1.08i)23-s + (−0.0999 − 0.173i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0977 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0977 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(285\)    =    \(3 \cdot 5 \cdot 19\)
Sign: $-0.0977 - 0.995i$
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: \((2,\ 285,\ (\ :1/2),\ -0.0977 - 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.838619 + 0.925011i\)
\(L(\frac12)\) \(\approx\) \(0.838619 + 0.925011i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.5 + 2.59i)T \)
good2 \( 1 + (-1 - 1.73i)T^{2} \)
7 \( 1 - 2T + 7T^{2} \)
11 \( 1 + 3T + 11T^{2} \)
13 \( 1 + (-2 - 3.46i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-3 - 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.5 + 2.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 5T + 31T^{2} \)
37 \( 1 - 8T + 37T^{2} \)
41 \( 1 + (-3 + 5.19i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2 + 3.46i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3 + 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3 - 5.19i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.5 + 7.79i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.5 - 6.06i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1 + 1.73i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-4.5 + 7.79i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-2 + 3.46i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.5 + 6.06i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (1.5 + 2.59i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5 + 8.66i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.70522320429231655425850609173, −11.28145519946896237305168221861, −10.51313632782297576579580533779, −9.137219930994872527620081041512, −8.165019330100892285752731708959, −7.29335768840567224342910229300, −6.21324663936253785246842856423, −4.78455255837517191829356593255, −3.73356401942851294073006599473, −2.34351914499770241363312561096, 1.03780072709899155447212920056, 2.56288290638956769447879286672, 4.65506023736418719204204806910, 5.57721125838198067926470722400, 6.50115687655823481505371873564, 7.79571012367595495667848250977, 8.434217602750645763792296431681, 9.946350406778203688306535106703, 10.84747121882814826328083154057, 11.34438472590641175599383608922

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