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Properties

Label 2-460-460.379-c1-0-11
Degree $2$
Conductor $460$
Sign $-0.590 - 0.807i$
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 + (−0.0739 + 0.514i)3-s + (0.830 − 1.81i)4-s + (0.629 + 2.14i)5-s + (−0.305 − 0.668i)6-s + (−0.858 + 0.743i)7-s + (0.402 + 2.79i)8-s + (2.61 + 0.769i)9-s + (−2.38 − 2.07i)10-s + (0.874 + 0.562i)12-s + (0.452 − 1.54i)14-s + (−1.15 + 0.165i)15-s + (−2.61 − 3.02i)16-s + (−3.70 + 1.08i)18-s + (4.42 + 0.636i)20-s + (−0.319 − 0.496i)21-s + ⋯
L(s)  = 1  + (−0.841 + 0.540i)2-s + (−0.0427 + 0.297i)3-s + (0.415 − 0.909i)4-s + (0.281 + 0.959i)5-s + (−0.124 − 0.273i)6-s + (−0.324 + 0.281i)7-s + (0.142 + 0.989i)8-s + (0.873 + 0.256i)9-s + (−0.755 − 0.654i)10-s + (0.252 + 0.162i)12-s + (0.120 − 0.411i)14-s + (−0.297 + 0.0427i)15-s + (−0.654 − 0.755i)16-s + (−0.873 + 0.256i)18-s + (0.989 + 0.142i)20-s + (−0.0696 − 0.108i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.590 - 0.807i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.590 - 0.807i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $-0.590 - 0.807i$
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: \((2,\ 460,\ (\ :1/2),\ -0.590 - 0.807i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.412310 + 0.812458i\)
\(L(\frac12)\) \(\approx\) \(0.412310 + 0.812458i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.96 - 4.37i)T \)
good3 \( 1 + (0.0739 - 0.514i)T + (-2.87 - 0.845i)T^{2} \)
7 \( 1 + (0.858 - 0.743i)T + (0.996 - 6.92i)T^{2} \)
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 \( 1 + (-1.34 - 2.94i)T + (-18.9 + 21.9i)T^{2} \)
31 \( 1 + (29.7 - 8.73i)T^{2} \)
37 \( 1 + (31.1 + 20.0i)T^{2} \)
41 \( 1 + (10.7 - 3.16i)T + (34.4 - 22.1i)T^{2} \)
43 \( 1 + (10.1 + 1.46i)T + (41.2 + 12.1i)T^{2} \)
47 \( 1 - 12.1T + 47T^{2} \)
53 \( 1 + (-7.54 + 52.4i)T^{2} \)
59 \( 1 + (8.39 + 58.3i)T^{2} \)
61 \( 1 + (1.68 - 0.242i)T + (58.5 - 17.1i)T^{2} \)
67 \( 1 + (1.05 + 1.63i)T + (-27.8 + 60.9i)T^{2} \)
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 \( 1 + (-3.95 + 13.4i)T + (-69.8 - 44.8i)T^{2} \)
89 \( 1 + (-18.1 - 2.60i)T + (85.3 + 25.0i)T^{2} \)
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

−10.99633645217624329266117050609, −10.26357102432154968756087289147, −9.698054605095510224348170365736, −8.774305501365847531176382257687, −7.55229820440233798606477675383, −6.92037052460183626157315804006, −5.99908466368425622122682041361, −4.91890749295253256201539893900, −3.26209204730741820977849623125, −1.79344461725355962982943673817, 0.77391220169484806821532313635, 2.04825656393189848109508527633, 3.67097272596367412455215840828, 4.77797090612995954351033772726, 6.35290201472455824809435780297, 7.19414881879124615342238299018, 8.239239761185904556502974814618, 8.996780997234042592636849352718, 9.908910394500302966534833100735, 10.44767936463802854956683672296

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