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Properties

Label 2-465-465.134-c0-0-0
Degree $2$
Conductor $465$
Sign $0.669 - 0.742i$
Analytic cond. $0.232065$
Root an. cond. $0.481731$
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  + (−1.58 − 1.14i)2-s + (0.104 + 0.994i)3-s + (0.873 + 2.68i)4-s + (0.5 + 0.866i)5-s + (0.978 − 1.69i)6-s + (1.10 − 3.39i)8-s + (−0.978 + 0.207i)9-s + (0.204 − 1.94i)10-s + (−2.58 + 1.14i)12-s + (−0.809 + 0.587i)15-s + (−3.36 + 2.44i)16-s + (0.669 + 0.743i)17-s + (1.78 + 0.795i)18-s + (−0.190 + 0.0850i)19-s + (−1.89 + 2.10i)20-s + ⋯
L(s)  = 1  + (−1.58 − 1.14i)2-s + (0.104 + 0.994i)3-s + (0.873 + 2.68i)4-s + (0.5 + 0.866i)5-s + (0.978 − 1.69i)6-s + (1.10 − 3.39i)8-s + (−0.978 + 0.207i)9-s + (0.204 − 1.94i)10-s + (−2.58 + 1.14i)12-s + (−0.809 + 0.587i)15-s + (−3.36 + 2.44i)16-s + (0.669 + 0.743i)17-s + (1.78 + 0.795i)18-s + (−0.190 + 0.0850i)19-s + (−1.89 + 2.10i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.669 - 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.669 - 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(465\)    =    \(3 \cdot 5 \cdot 31\)
Sign: $0.669 - 0.742i$
Analytic conductor: \(0.232065\)
Root analytic conductor: \(0.481731\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{465} (134, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 465,\ (\ :0),\ 0.669 - 0.742i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4397702439\)
\(L(\frac12)\) \(\approx\) \(0.4397702439\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.104 - 0.994i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 + (0.809 + 0.587i)T \)
good2 \( 1 + (1.58 + 1.14i)T + (0.309 + 0.951i)T^{2} \)
7 \( 1 + (-0.913 - 0.406i)T^{2} \)
11 \( 1 + (0.104 + 0.994i)T^{2} \)
13 \( 1 + (-0.669 - 0.743i)T^{2} \)
17 \( 1 + (-0.669 - 0.743i)T + (-0.104 + 0.994i)T^{2} \)
19 \( 1 + (0.190 - 0.0850i)T + (0.669 - 0.743i)T^{2} \)
23 \( 1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2} \)
29 \( 1 + (-0.309 - 0.951i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 + (0.978 + 0.207i)T^{2} \)
43 \( 1 + (-0.669 + 0.743i)T^{2} \)
47 \( 1 + (-1.08 + 0.786i)T + (0.309 - 0.951i)T^{2} \)
53 \( 1 + (1.58 - 0.336i)T + (0.913 - 0.406i)T^{2} \)
59 \( 1 + (0.978 - 0.207i)T^{2} \)
61 \( 1 - 1.82T + T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.913 + 0.406i)T^{2} \)
73 \( 1 + (0.104 + 0.994i)T^{2} \)
79 \( 1 + (0.139 + 0.155i)T + (-0.104 + 0.994i)T^{2} \)
83 \( 1 + (-0.139 + 1.33i)T + (-0.978 - 0.207i)T^{2} \)
89 \( 1 + (0.809 - 0.587i)T^{2} \)
97 \( 1 + (0.809 - 0.587i)T^{2} \)
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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

−10.98899178039223208632675436189, −10.36663140750130719700645811460, −9.780596432333432252044333299637, −9.070595795789156853826406961097, −8.131931306410381535360510082717, −7.22562218196182968057295857469, −5.87668394911897982044467704037, −3.95885014133805762819881333009, −3.14138438729138123947847482513, −2.01598465789310188260066149448, 0.979127647107487759404565963560, 2.22266823897879628221397964537, 5.11240298815598550152785122659, 5.88152619076816721288308716444, 6.77734389552302565570464003949, 7.61364275240417034321289125042, 8.400054213252314964512756336396, 9.055152941732057387989482058193, 9.809105340772739027389757863988, 10.84331441985507508057376185063

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