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Properties

Label 2-465-465.329-c0-0-1
Degree $2$
Conductor $465$
Sign $0.987 - 0.157i$
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  + (1.08 + 0.786i)2-s + (−0.913 − 0.406i)3-s + (0.244 + 0.752i)4-s + (0.5 − 0.866i)5-s + (−0.669 − 1.15i)6-s + (0.0864 − 0.266i)8-s + (0.669 + 0.743i)9-s + (1.22 − 0.544i)10-s + (0.0826 − 0.786i)12-s + (−0.809 + 0.587i)15-s + (0.942 − 0.684i)16-s + (−0.978 + 0.207i)17-s + (0.139 + 1.33i)18-s + (−0.190 + 1.81i)19-s + (0.773 + 0.164i)20-s + ⋯
L(s)  = 1  + (1.08 + 0.786i)2-s + (−0.913 − 0.406i)3-s + (0.244 + 0.752i)4-s + (0.5 − 0.866i)5-s + (−0.669 − 1.15i)6-s + (0.0864 − 0.266i)8-s + (0.669 + 0.743i)9-s + (1.22 − 0.544i)10-s + (0.0826 − 0.786i)12-s + (−0.809 + 0.587i)15-s + (0.942 − 0.684i)16-s + (−0.978 + 0.207i)17-s + (0.139 + 1.33i)18-s + (−0.190 + 1.81i)19-s + (0.773 + 0.164i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.157i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.157i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(465\)    =    \(3 \cdot 5 \cdot 31\)
Sign: $0.987 - 0.157i$
Analytic conductor: \(0.232065\)
Root analytic conductor: \(0.481731\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{465} (329, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 465,\ (\ :0),\ 0.987 - 0.157i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.194889153\)
\(L(\frac12)\) \(\approx\) \(1.194889153\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.913 + 0.406i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 + (0.809 + 0.587i)T \)
good2 \( 1 + (-1.08 - 0.786i)T + (0.309 + 0.951i)T^{2} \)
7 \( 1 + (0.104 + 0.994i)T^{2} \)
11 \( 1 + (-0.913 - 0.406i)T^{2} \)
13 \( 1 + (0.978 - 0.207i)T^{2} \)
17 \( 1 + (0.978 - 0.207i)T + (0.913 - 0.406i)T^{2} \)
19 \( 1 + (0.190 - 1.81i)T + (-0.978 - 0.207i)T^{2} \)
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 \( 1 + (-0.669 + 0.743i)T^{2} \)
43 \( 1 + (0.978 + 0.207i)T^{2} \)
47 \( 1 + (1.58 - 1.14i)T + (0.309 - 0.951i)T^{2} \)
53 \( 1 + (-1.08 - 1.20i)T + (-0.104 + 0.994i)T^{2} \)
59 \( 1 + (-0.669 - 0.743i)T^{2} \)
61 \( 1 + 0.209T + T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.104 - 0.994i)T^{2} \)
73 \( 1 + (-0.913 - 0.406i)T^{2} \)
79 \( 1 + (1.78 - 0.379i)T + (0.913 - 0.406i)T^{2} \)
83 \( 1 + (-1.78 + 0.795i)T + (0.669 - 0.743i)T^{2} \)
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

−11.63570718447560590455593054175, −10.43611306507586904500977451168, −9.601427306325979108182134730530, −8.243905033263093650077685102374, −7.30651982646699772333777696869, −6.18170843677972985748785242276, −5.76795933830218760517433682626, −4.82764859226951970867722484627, −3.96048637845052247290497248046, −1.68207407275644732539480900222, 2.19420369782610027285478852038, 3.35485337814215447136528948385, 4.51729630872309131675704942683, 5.26660824581268860261745043245, 6.36110461392365246096419286808, 7.08477525696123178811592694959, 8.810151903571285313779105242599, 9.900603283867434544718914202200, 10.82109638118087509002080001504, 11.17828552217020153307491345678

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