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Properties

Label 2-495-495.32-c0-0-1
Degree $2$
Conductor $495$
Sign $0.949 + 0.313i$
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 + (−1.36 + 0.366i)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 + (−1.36 − 1.36i)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 + (−1.36 + 0.366i)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 + (−1.36 − 1.36i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.949 + 0.313i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.949 + 0.313i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(495\)    =    \(3^{2} \cdot 5 \cdot 11\)
Sign: $0.949 + 0.313i$
Analytic conductor: \(0.247037\)
Root analytic conductor: \(0.497028\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{495} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 495,\ (\ :0),\ 0.949 + 0.313i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.065284270\)
\(L(\frac12)\) \(\approx\) \(1.065284270\)
\(L(1)\) 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 \( 1 + (1.36 - 0.366i)T + (0.866 - 0.5i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (1.36 + 1.36i)T + iT^{2} \)
41 \( 1 + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.866 - 0.5i)T^{2} \)
47 \( 1 + (0.5 + 0.133i)T + (0.866 + 0.5i)T^{2} \)
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 \( 1 + (-0.5 + 0.133i)T + (0.866 - 0.5i)T^{2} \)
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 \( 1 + (0.133 - 0.5i)T + (-0.866 - 0.5i)T^{2} \)
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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

−11.32938173185028616485613662592, −10.49899360033433858244368815618, −8.996942436791592701200212598180, −8.246681449632261562896239370392, −7.39526706422629337891072562198, −6.82607524952215283666110859431, −5.91404957157378762154368505152, −3.84628863297843511528882371314, −3.20088019849690015011782082664, −1.84721426578246701419193125510, 1.95870218739360182364568451471, 3.46686617778908616612350240924, 4.38030587005669290651723648087, 5.44868904860580326033723017057, 6.66931660313744646780275488795, 7.72168927486894143644440710111, 8.527666667096596938007767975745, 9.589694155967324410893165034320, 10.19894423648109742788101761137, 11.32994000838369861258506317715

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