Properties

Label 2-460-460.379-c1-0-46
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

Related objects

Downloads

Learn more

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

\(L(1)\) \(\approx\) \(2.24087 - 1.13720i\)
\(L(\frac12)\) \(\approx\) \(2.24087 - 1.13720i\)
\(L(\frac{3}{2})\) 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 \( 1 + (81.6 - 52.4i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.88009606073341413872872162313, −10.35189094161898332197240033265, −9.532030754129151829667633323798, −7.956815557661026853795652459469, −6.93702668895215383406099523003, −6.31828635259390837044420781636, −5.02546505188849485522449680522, −3.99599475572883090520940705705, −2.76682444915229589528480663553, −1.59692697173169484440060049114, 1.85788258497648241401037633719, 3.59407864199093691636243985401, 4.58420737662030096688351529479, 5.33364906905058084106635776153, 6.34675973258352564355116248732, 7.48464462311206619751626544654, 8.376597264671417214558632120923, 9.271967703297241601517580244925, 10.22712909940495160140670441016, 11.54970928368327621725737342119

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