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Properties

Label 2-465-465.59-c0-0-1
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: $\chi_{465} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 465,\ (\ :0),\ 0.669 + 0.742i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.645938622\)
\(L(\frac12)\) \(\approx\) \(1.645938622\)
\(L(1)\) 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

−11.30145937449428419510759138334, −10.51084791208460051079135568873, −10.03272968203044817609501089645, −8.806340262205371895131893214657, −6.99426960417673635958891303785, −6.02859973681540381863740226866, −5.09505852012124105691851996524, −4.03076614863049817693286838980, −3.46579053368725549075210533157, −2.32113414567593505922990885568, 2.46879601323785015930959832518, 3.86350197024384300628364228715, 4.90442813395917164895037239582, 5.68990683386446654336576740095, 6.70944954809322348130375846666, 7.41419763043861717411486988454, 8.233777866368959228250218388100, 8.956110639096674668815073116246, 11.22845596830022499484331001051, 11.78118336648139734673152139406

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