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Properties

Label 2-465-465.224-c0-0-1
Degree $2$
Conductor $465$
Sign $-0.807 + 0.589i$
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  + (−0.564 − 1.73i)2-s + (0.978 − 0.207i)3-s + (−1.89 + 1.37i)4-s + (0.5 − 0.866i)5-s + (−0.913 − 1.58i)6-s + (1.97 + 1.43i)8-s + (0.913 − 0.406i)9-s + (−1.78 − 0.379i)10-s + (−1.56 + 1.73i)12-s + (0.309 − 0.951i)15-s + (0.657 − 2.02i)16-s + (−0.104 + 0.994i)17-s + (−1.22 − 1.35i)18-s + (−1.30 + 1.45i)19-s + (0.244 + 2.32i)20-s + ⋯
L(s)  = 1  + (−0.564 − 1.73i)2-s + (0.978 − 0.207i)3-s + (−1.89 + 1.37i)4-s + (0.5 − 0.866i)5-s + (−0.913 − 1.58i)6-s + (1.97 + 1.43i)8-s + (0.913 − 0.406i)9-s + (−1.78 − 0.379i)10-s + (−1.56 + 1.73i)12-s + (0.309 − 0.951i)15-s + (0.657 − 2.02i)16-s + (−0.104 + 0.994i)17-s + (−1.22 − 1.35i)18-s + (−1.30 + 1.45i)19-s + (0.244 + 2.32i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.807 + 0.589i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.807 + 0.589i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(465\)    =    \(3 \cdot 5 \cdot 31\)
Sign: $-0.807 + 0.589i$
Analytic conductor: \(0.232065\)
Root analytic conductor: \(0.481731\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{465} (224, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 465,\ (\ :0),\ -0.807 + 0.589i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8512452116\)
\(L(\frac12)\) \(\approx\) \(0.8512452116\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.978 + 0.207i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 + (-0.309 - 0.951i)T \)
good2 \( 1 + (0.564 + 1.73i)T + (-0.809 + 0.587i)T^{2} \)
7 \( 1 + (-0.669 - 0.743i)T^{2} \)
11 \( 1 + (0.978 - 0.207i)T^{2} \)
13 \( 1 + (0.104 - 0.994i)T^{2} \)
17 \( 1 + (0.104 - 0.994i)T + (-0.978 - 0.207i)T^{2} \)
19 \( 1 + (1.30 - 1.45i)T + (-0.104 - 0.994i)T^{2} \)
23 \( 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)T^{2} \)
29 \( 1 + (0.809 - 0.587i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.913 - 0.406i)T^{2} \)
43 \( 1 + (0.104 + 0.994i)T^{2} \)
47 \( 1 + (-0.0646 + 0.198i)T + (-0.809 - 0.587i)T^{2} \)
53 \( 1 + (0.564 - 0.251i)T + (0.669 - 0.743i)T^{2} \)
59 \( 1 + (-0.913 + 0.406i)T^{2} \)
61 \( 1 - 1.33T + T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.669 + 0.743i)T^{2} \)
73 \( 1 + (0.978 - 0.207i)T^{2} \)
79 \( 1 + (-0.204 + 1.94i)T + (-0.978 - 0.207i)T^{2} \)
83 \( 1 + (0.204 + 0.0434i)T + (0.913 + 0.406i)T^{2} \)
89 \( 1 + (-0.309 + 0.951i)T^{2} \)
97 \( 1 + (-0.309 + 0.951i)T^{2} \)
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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

−10.45516902458461774172417097649, −10.21061161973832015572766633449, −9.188729417711295512716811164589, −8.443978150071307125080333628466, −8.059515437068500862489750220573, −6.20216752767361646757679751164, −4.44755270965318884367978486224, −3.75548932802332619620886991724, −2.31613845537286452056650186894, −1.54647328164570292722048344978, 2.40866207458106990006195841559, 4.07784348472268510782985739312, 5.25742250890246169110994947315, 6.41999012637080455826712328971, 7.10364997920070288779919919962, 7.87786352392975329356587695111, 8.789670798470460792323326417210, 9.561977614440646582839685244613, 10.11641276188211851364413516505, 11.28050085973178033868534486084

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