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Properties

Label 2-285-19.7-c1-0-11
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 − 1.73i)2-s + (−0.5 − 0.866i)3-s + (−0.999 + 1.73i)4-s + (0.5 + 0.866i)5-s + (−0.999 + 1.73i)6-s − 2·7-s + (−0.499 + 0.866i)9-s + (0.999 − 1.73i)10-s − 3·11-s + 1.99·12-s + (−3 + 5.19i)13-s + (2 + 3.46i)14-s + (0.499 − 0.866i)15-s + (1.99 + 3.46i)16-s + (−3 − 5.19i)17-s + 1.99·18-s + ⋯
L(s)  = 1  + (−0.707 − 1.22i)2-s + (−0.288 − 0.499i)3-s + (−0.499 + 0.866i)4-s + (0.223 + 0.387i)5-s + (−0.408 + 0.707i)6-s − 0.755·7-s + (−0.166 + 0.288i)9-s + (0.316 − 0.547i)10-s − 0.904·11-s + 0.577·12-s + (−0.832 + 1.44i)13-s + (0.534 + 0.925i)14-s + (0.129 − 0.223i)15-s + (0.499 + 0.866i)16-s + (−0.727 − 1.26i)17-s + 0.471·18-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: $\chi_{285} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 285,\ (\ :1/2),\ 0.0977 - 0.995i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 + 1.73i)T + (-1 + 1.73i)T^{2} \)
7 \( 1 + 2T + 7T^{2} \)
11 \( 1 + 3T + 11T^{2} \)
13 \( 1 + (3 - 5.19i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (3 + 5.19i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-4 + 6.92i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.5 - 6.06i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 9T + 31T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 + (3 + 5.19i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (5 + 8.66i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2 - 3.46i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (7 - 12.1i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.5 + 2.59i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.5 + 6.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2 - 3.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.5 + 6.06i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (1 + 1.73i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.5 + 4.33i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 6T + 83T^{2} \)
89 \( 1 + (1.5 - 2.59i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-6 - 10.3i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.98736794779768087415666667552, −10.32776115212624264008329999657, −9.423290489395488255228182447854, −8.601920224152026676854597448667, −7.11203936558606311489231782150, −6.38110975283015426625228134665, −4.74765612346183049747446150585, −2.96082758001508475377852847490, −2.08658067997107687672058176598, 0, 3.02371113009014549865396421030, 4.85758787905193926360500455544, 5.78268758879400743688009078343, 6.63694526306519805057173474227, 7.86714648183370700147441033604, 8.544249894031179165270497702281, 9.754558477423477865376608101697, 10.13064529512781838969351215215, 11.44041867504039776305930414875

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