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Properties

Label 2-495-495.362-c0-0-1
Degree $2$
Conductor $495$
Sign $0.929 - 0.370i$
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 + (−0.366 + 1.36i)23-s − 25-s − 0.999i·27-s + (−0.866 − 1.5i)31-s + 0.999·33-s + 0.999i·36-s + (−1.36 − 1.36i)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 + (−0.366 + 1.36i)23-s − 25-s − 0.999i·27-s + (−0.866 − 1.5i)31-s + 0.999·33-s + 0.999i·36-s + (−1.36 − 1.36i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 - 0.370i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 - 0.370i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(495\)    =    \(3^{2} \cdot 5 \cdot 11\)
Sign: $0.929 - 0.370i$
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: \((2,\ 495,\ (\ :0),\ 0.929 - 0.370i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9843008209\)
\(L(\frac12)\) \(\approx\) \(0.9843008209\)
\(L(1)\) 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 \( 1 + (0.366 - 1.36i)T + (-0.866 - 0.5i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (1.36 + 1.36i)T + iT^{2} \)
41 \( 1 + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.866 - 0.5i)T^{2} \)
47 \( 1 + (0.133 + 0.5i)T + (-0.866 + 0.5i)T^{2} \)
53 \( 1 + (1.36 + 1.36i)T + iT^{2} \)
59 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.133 - 0.5i)T + (-0.866 - 0.5i)T^{2} \)
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 \( 1 + (-0.5 + 0.133i)T + (0.866 - 0.5i)T^{2} \)
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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

−11.38621769151760262933131791566, −9.971460437559183772952906444944, −9.424699080940727596072468759345, −8.534976750902787077979997420767, −7.50885054841182019843735274232, −7.02766258505508733055543602208, −5.70737179488787355466356858827, −4.02244641940602479115619138028, −3.47334679417352527692834510705, −2.05262932144264499254348543129, 1.52899986239110933751859444128, 3.43864668090932574591959192823, 4.44201124495731716410899027414, 5.11636428673202979776944937777, 6.36981492368732319401894874691, 7.910691812523036008691482808305, 8.831882657656863280690989418539, 9.011256702419911035230060266649, 10.03525184284950303808943253044, 10.80414587856532024422099097628

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