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Properties

Label 2-495-495.362-c0-0-0
Degree $2$
Conductor $495$
Sign $0.144 - 0.989i$
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  + (0.5 + 0.866i)3-s + (−0.866 + 0.5i)4-s + (0.866 + 0.5i)5-s + (−0.499 + 0.866i)9-s + (−0.866 − 0.5i)11-s + (−0.866 − 0.499i)12-s + 0.999i·15-s + (0.499 − 0.866i)16-s − 0.999·20-s + (0.366 − 1.36i)23-s + (0.499 + 0.866i)25-s − 0.999·27-s + (0.866 + 1.5i)31-s − 0.999i·33-s − 0.999i·36-s + (0.366 + 0.366i)37-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)3-s + (−0.866 + 0.5i)4-s + (0.866 + 0.5i)5-s + (−0.499 + 0.866i)9-s + (−0.866 − 0.5i)11-s + (−0.866 − 0.499i)12-s + 0.999i·15-s + (0.499 − 0.866i)16-s − 0.999·20-s + (0.366 − 1.36i)23-s + (0.499 + 0.866i)25-s − 0.999·27-s + (0.866 + 1.5i)31-s − 0.999i·33-s − 0.999i·36-s + (0.366 + 0.366i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.144 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.144 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(495\)    =    \(3^{2} \cdot 5 \cdot 11\)
Sign: $0.144 - 0.989i$
Analytic conductor: \(0.247037\)
Root analytic conductor: \(0.497028\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{495} (362, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 495,\ (\ :0),\ 0.144 - 0.989i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9237486790\)
\(L(\frac12)\) \(\approx\) \(0.9237486790\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + (-0.866 - 0.5i)T \)
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 \( 1 + (-0.366 + 1.36i)T + (-0.866 - 0.5i)T^{2} \)
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 \( 1 + (0.5 + 1.86i)T + (-0.866 + 0.5i)T^{2} \)
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 \( 1 + (-0.5 + 1.86i)T + (-0.866 - 0.5i)T^{2} \)
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 \( 1 + (1.86 - 0.5i)T + (0.866 - 0.5i)T^{2} \)
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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

−11.01054552545959981124645862362, −10.29790529887516113091830294552, −9.664152664443374442204985011561, −8.660737014743874178137162387988, −8.181097224403250076429534382984, −6.80899738404573526856472867657, −5.44255216402465536099712575134, −4.74408989759546288632745256594, −3.43421593984632604811073587241, −2.58556990613580809599912364516, 1.33564231201802237818493610602, 2.65257129196170521340482731282, 4.32308794592316685357890311148, 5.46479269756150405194829372898, 6.14922570323653011312389065793, 7.50602296090919361362358649251, 8.291844440638076702946115553632, 9.357647366807452098214582655108, 9.667965732094425195470625002070, 10.83345382211705143744275765905

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