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Properties

Label 2-495-495.263-c0-0-1
Degree $2$
Conductor $495$
Sign $0.746 + 0.665i$
Analytic cond. $0.247037$
Root an. cond. $0.497028$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 + 0.5i)3-s + (0.866 − 0.5i)4-s − i·5-s + (0.499 − 0.866i)9-s + (−0.866 − 0.5i)11-s + (−0.499 + 0.866i)12-s + (0.5 + 0.866i)15-s + (0.499 − 0.866i)16-s + (−0.5 − 0.866i)20-s + (1.36 + 0.366i)23-s − 25-s + 0.999i·27-s + (0.866 + 1.5i)31-s + 0.999·33-s − 0.999i·36-s + (0.366 − 0.366i)37-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)3-s + (0.866 − 0.5i)4-s − i·5-s + (0.499 − 0.866i)9-s + (−0.866 − 0.5i)11-s + (−0.499 + 0.866i)12-s + (0.5 + 0.866i)15-s + (0.499 − 0.866i)16-s + (−0.5 − 0.866i)20-s + (1.36 + 0.366i)23-s − 25-s + 0.999i·27-s + (0.866 + 1.5i)31-s + 0.999·33-s − 0.999i·36-s + (0.366 − 0.366i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.746 + 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.746 + 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(495\)    =    \(3^{2} \cdot 5 \cdot 11\)
Sign: $0.746 + 0.665i$
Analytic conductor: \(0.247037\)
Root analytic conductor: \(0.497028\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{495} (263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 495,\ (\ :0),\ 0.746 + 0.665i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7909000775\)
\(L(\frac12)\) \(\approx\) \(0.7909000775\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.866 - 0.5i)T \)
5 \( 1 + iT \)
11 \( 1 + (0.866 + 0.5i)T \)
good2 \( 1 + (-0.866 + 0.5i)T^{2} \)
7 \( 1 + (0.866 - 0.5i)T^{2} \)
13 \( 1 + (0.866 + 0.5i)T^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + (-1.36 - 0.366i)T + (0.866 + 0.5i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.366 + 0.366i)T - iT^{2} \)
41 \( 1 + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.866 + 0.5i)T^{2} \)
47 \( 1 + (1.86 - 0.5i)T + (0.866 - 0.5i)T^{2} \)
53 \( 1 + (-0.366 + 0.366i)T - iT^{2} \)
59 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (1.86 + 0.5i)T + (0.866 + 0.5i)T^{2} \)
71 \( 1 - 1.73iT - T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.866 - 0.5i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (-0.5 - 1.86i)T + (-0.866 + 0.5i)T^{2} \)
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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

−11.07296355807486084267396404534, −10.31255438674692668626928065955, −9.499936602162088523128288025302, −8.449927362760735828764032629997, −7.26303785786482097245972056960, −6.23260338478980074388548502978, −5.36265622384288848972000656671, −4.73364441736028682641559194240, −3.10598679305666000735317067272, −1.23492128669739442889899440459, 2.08029747378175582595134055297, 3.09958121648173750231715939078, 4.71582593023957418227651834504, 5.99523151051759443651977132911, 6.72408961889260864726577211928, 7.44442410498619269434109463450, 8.138274389718141439876325593246, 9.892269155656133594933242152304, 10.61823281822263638790643081830, 11.32359911362484093318875105356

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