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Properties

Label 2-460-460.379-c1-0-51
Degree $2$
Conductor $460$
Sign $0.800 + 0.599i$
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  + (1.18 − 0.764i)2-s + (−0.412 + 2.86i)3-s + (0.830 − 1.81i)4-s + (−0.629 − 2.14i)5-s + (1.70 + 3.72i)6-s + (3.98 − 3.45i)7-s + (−0.402 − 2.79i)8-s + (−5.18 − 1.52i)9-s + (−2.38 − 2.07i)10-s + (4.87 + 3.13i)12-s + (2.10 − 7.16i)14-s + (6.41 − 0.922i)15-s + (−2.61 − 3.02i)16-s + (−7.33 + 2.15i)18-s + (−4.42 − 0.636i)20-s + (8.27 + 12.8i)21-s + ⋯
L(s)  = 1  + (0.841 − 0.540i)2-s + (−0.238 + 1.65i)3-s + (0.415 − 0.909i)4-s + (−0.281 − 0.959i)5-s + (0.695 + 1.52i)6-s + (1.50 − 1.30i)7-s + (−0.142 − 0.989i)8-s + (−1.72 − 0.507i)9-s + (−0.755 − 0.654i)10-s + (1.40 + 0.905i)12-s + (0.561 − 1.91i)14-s + (1.65 − 0.238i)15-s + (−0.654 − 0.755i)16-s + (−1.72 + 0.507i)18-s + (−0.989 − 0.142i)20-s + (1.80 + 2.80i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.800 + 0.599i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.800 + 0.599i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $0.800 + 0.599i$
Analytic conductor: \(3.67311\)
Root analytic conductor: \(1.91653\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 460,\ (\ :1/2),\ 0.800 + 0.599i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.08604 - 0.695085i\)
\(L(\frac12)\) \(\approx\) \(2.08604 - 0.695085i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.18 + 0.764i)T \)
5 \( 1 + (0.629 + 2.14i)T \)
23 \( 1 + (-3.15 - 3.60i)T \)
good3 \( 1 + (0.412 - 2.86i)T + (-2.87 - 0.845i)T^{2} \)
7 \( 1 + (-3.98 + 3.45i)T + (0.996 - 6.92i)T^{2} \)
11 \( 1 + (4.56 + 10.0i)T^{2} \)
13 \( 1 + (1.85 + 12.8i)T^{2} \)
17 \( 1 + (-11.1 + 12.8i)T^{2} \)
19 \( 1 + (-12.4 - 14.3i)T^{2} \)
29 \( 1 + (-3.43 - 7.52i)T + (-18.9 + 21.9i)T^{2} \)
31 \( 1 + (29.7 - 8.73i)T^{2} \)
37 \( 1 + (31.1 + 20.0i)T^{2} \)
41 \( 1 + (4.29 - 1.26i)T + (34.4 - 22.1i)T^{2} \)
43 \( 1 + (-12.7 - 1.83i)T + (41.2 + 12.1i)T^{2} \)
47 \( 1 + 6.82T + 47T^{2} \)
53 \( 1 + (-7.54 + 52.4i)T^{2} \)
59 \( 1 + (8.39 + 58.3i)T^{2} \)
61 \( 1 + (12.7 - 1.82i)T + (58.5 - 17.1i)T^{2} \)
67 \( 1 + (-3.48 - 5.42i)T + (-27.8 + 60.9i)T^{2} \)
71 \( 1 + (-29.4 + 64.5i)T^{2} \)
73 \( 1 + (47.8 + 55.1i)T^{2} \)
79 \( 1 + (-11.2 - 78.1i)T^{2} \)
83 \( 1 + (4.71 - 16.0i)T + (-69.8 - 44.8i)T^{2} \)
89 \( 1 + (11.6 + 1.68i)T + (85.3 + 25.0i)T^{2} \)
97 \( 1 + (81.6 - 52.4i)T^{2} \)
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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

−11.05693765280900307967070584485, −10.36398157759422232174118484528, −9.477747827246180500999860162120, −8.494995645345605338239623692745, −7.26965126243259078855538557615, −5.53981607073926935894501643632, −4.80054352796094694374935153752, −4.37381399189950521012212222127, −3.45645801556940095450133028978, −1.26039410339214195230356478878, 2.01807455005146475095546722765, 2.78331897168977940286831962819, 4.65657392977861527682394236001, 5.80492666099283101651493000187, 6.40304647277706305331323694696, 7.44528618111914260962430133606, 7.979244997462419570478542234814, 8.747390219647192171599320729852, 10.88470051180712343125681085632, 11.57222602753555095650622467967

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