Properties

Label 2-460-460.339-c1-0-26
Degree $2$
Conductor $460$
Sign $-0.408 - 0.912i$
Analytic cond. $3.67311$
Root an. cond. $1.91653$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.587 + 1.28i)2-s + (3.32 + 0.975i)3-s + (−1.30 − 1.51i)4-s + (1.20 + 1.88i)5-s + (−3.20 + 3.70i)6-s + (−3.71 − 0.533i)7-s + (2.71 − 0.796i)8-s + (7.57 + 4.86i)9-s + (−3.13 + 0.450i)10-s + (−2.87 − 6.30i)12-s + (2.86 − 4.46i)14-s + (2.18 + 7.43i)15-s + (−0.569 + 3.95i)16-s + (−10.7 + 6.88i)18-s + (1.25 − 4.29i)20-s + (−11.8 − 5.39i)21-s + ⋯
L(s)  = 1  + (−0.415 + 0.909i)2-s + (1.91 + 0.563i)3-s + (−0.654 − 0.755i)4-s + (0.540 + 0.841i)5-s + (−1.30 + 1.51i)6-s + (−1.40 − 0.201i)7-s + (0.959 − 0.281i)8-s + (2.52 + 1.62i)9-s + (−0.989 + 0.142i)10-s + (−0.830 − 1.81i)12-s + (0.766 − 1.19i)14-s + (0.563 + 1.91i)15-s + (−0.142 + 0.989i)16-s + (−2.52 + 1.62i)18-s + (0.281 − 0.959i)20-s + (−2.57 − 1.17i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.408 - 0.912i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.408 - 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $-0.408 - 0.912i$
Analytic conductor: \(3.67311\)
Root analytic conductor: \(1.91653\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (339, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 460,\ (\ :1/2),\ -0.408 - 0.912i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.06593 + 1.64394i\)
\(L(\frac12)\) \(\approx\) \(1.06593 + 1.64394i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.587 - 1.28i)T \)
5 \( 1 + (-1.20 - 1.88i)T \)
23 \( 1 + (4.60 + 1.32i)T \)
good3 \( 1 + (-3.32 - 0.975i)T + (2.52 + 1.62i)T^{2} \)
7 \( 1 + (3.71 + 0.533i)T + (6.71 + 1.97i)T^{2} \)
11 \( 1 + (-7.20 + 8.31i)T^{2} \)
13 \( 1 + (12.4 - 3.66i)T^{2} \)
17 \( 1 + (-2.41 - 16.8i)T^{2} \)
19 \( 1 + (-2.70 + 18.8i)T^{2} \)
29 \( 1 + (-6.47 + 7.46i)T + (-4.12 - 28.7i)T^{2} \)
31 \( 1 + (-26.0 + 16.7i)T^{2} \)
37 \( 1 + (15.3 + 33.6i)T^{2} \)
41 \( 1 + (-5.16 + 3.31i)T + (17.0 - 37.2i)T^{2} \)
43 \( 1 + (-2.12 + 7.24i)T + (-36.1 - 23.2i)T^{2} \)
47 \( 1 + 3.19T + 47T^{2} \)
53 \( 1 + (-50.8 - 14.9i)T^{2} \)
59 \( 1 + (56.6 - 16.6i)T^{2} \)
61 \( 1 + (0.771 + 2.62i)T + (-51.3 + 32.9i)T^{2} \)
67 \( 1 + (12.7 + 5.80i)T + (43.8 + 50.6i)T^{2} \)
71 \( 1 + (46.4 + 53.6i)T^{2} \)
73 \( 1 + (10.3 - 72.2i)T^{2} \)
79 \( 1 + (-75.7 + 22.2i)T^{2} \)
83 \( 1 + (2.55 - 3.97i)T + (-34.4 - 75.4i)T^{2} \)
89 \( 1 + (-0.555 + 1.89i)T + (-74.8 - 48.1i)T^{2} \)
97 \( 1 + (40.2 - 88.2i)T^{2} \)
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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

−10.55884647236001352670140178299, −9.996894251409036615938161227934, −9.544680784685733611225107371608, −8.699243626204717998620960282335, −7.74421704272361631290026027543, −6.96229165223115466293364820662, −6.04975811811838756855341199012, −4.35621520812960033998130916422, −3.37520899858363367170646213088, −2.26144303917712098919368970417, 1.30918787973356173995576002940, 2.53984148474346936680287043159, 3.32624470105423015404899278941, 4.41707820165584707477772046710, 6.35392816751367315875840332601, 7.50842325838516821991492064927, 8.445371979942617883688391284248, 9.044550195748661279291763204833, 9.678390192721257341454948314666, 10.21768241596438729766941487747

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