↑

Properties

Label 1-496-496.245-r0-0-0
Degree $1$
Conductor $496$
Sign $0.334 - 0.942i$
Analytic cond. $2.30341$
Root an. cond. $2.30341$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.207 + 0.978i)3-s + (0.866 − 0.5i)5-s + (−0.913 + 0.406i)7-s + (−0.913 − 0.406i)9-s + (−0.994 − 0.104i)11-s + (−0.743 − 0.669i)13-s + (0.309 + 0.951i)15-s + (−0.104 − 0.994i)17-s + (0.743 − 0.669i)19-s + (−0.207 − 0.978i)21-s + (0.809 − 0.587i)23-s + (0.5 − 0.866i)25-s + (0.587 − 0.809i)27-s + (−0.951 − 0.309i)29-s + (0.309 − 0.951i)33-s + ⋯
L(s)  = 1  + (−0.207 + 0.978i)3-s + (0.866 − 0.5i)5-s + (−0.913 + 0.406i)7-s + (−0.913 − 0.406i)9-s + (−0.994 − 0.104i)11-s + (−0.743 − 0.669i)13-s + (0.309 + 0.951i)15-s + (−0.104 − 0.994i)17-s + (0.743 − 0.669i)19-s + (−0.207 − 0.978i)21-s + (0.809 − 0.587i)23-s + (0.5 − 0.866i)25-s + (0.587 − 0.809i)27-s + (−0.951 − 0.309i)29-s + (0.309 − 0.951i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 496 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.334 - 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 496 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.334 - 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(496\)    =    \(2^{4} \cdot 31\)
Sign: $0.334 - 0.942i$
Analytic conductor: \(2.30341\)
Root analytic conductor: \(2.30341\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{496} (245, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 496,\ (0:\ ),\ 0.334 - 0.942i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6202153509 - 0.4379929480i\)
\(L(\frac12)\) \(\approx\) \(0.6202153509 - 0.4379929480i\)
\(L(1)\) \(\approx\) \(0.8335046974 + 0.01205136732i\)
\(L(1)\) \(\approx\) \(0.8335046974 + 0.01205136732i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
31 \( 1 \)
good3 \( 1 + (-0.207 + 0.978i)T \)
5 \( 1 + (0.866 - 0.5i)T \)
7 \( 1 + (-0.913 + 0.406i)T \)
11 \( 1 + (-0.994 - 0.104i)T \)
13 \( 1 + (-0.743 - 0.669i)T \)
17 \( 1 + (-0.104 - 0.994i)T \)
19 \( 1 + (0.743 - 0.669i)T \)
23 \( 1 + (0.809 - 0.587i)T \)
29 \( 1 + (-0.951 - 0.309i)T \)
37 \( 1 + (-0.866 - 0.5i)T \)
41 \( 1 + (0.978 - 0.207i)T \)
43 \( 1 + (-0.743 + 0.669i)T \)
47 \( 1 + (0.309 + 0.951i)T \)
53 \( 1 + (-0.406 + 0.913i)T \)
59 \( 1 + (0.207 - 0.978i)T \)
61 \( 1 - iT \)
67 \( 1 + (-0.866 + 0.5i)T \)
71 \( 1 + (-0.913 - 0.406i)T \)
73 \( 1 + (0.104 - 0.994i)T \)
79 \( 1 + (-0.104 - 0.994i)T \)
83 \( 1 + (0.207 + 0.978i)T \)
89 \( 1 + (0.809 + 0.587i)T \)
97 \( 1 + (-0.809 - 0.587i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−23.86239316304848148747238995618, −22.995252014311620800002386782679, −22.33159363562303629004127992028, −21.42995987354556807151498777443, −20.39352817608957622861762231689, −19.326915789142450525449370758585, −18.80723728933862600041767069754, −17.952509151244507230525472950986, −17.10221158257744146375735311515, −16.47455370593819568170971605217, −15.11303838158265222451083425983, −14.16295857043742576063800028143, −13.30817977097977279549840274699, −12.86289673144899851481090490095, −11.779107818429783383971300699176, −10.62967268411425246561924488700, −9.94777653426980081219819048762, −8.8734810848413450531353575745, −7.494176827676912717932493099243, −6.975440879246222475832158281635, −5.984403779119715608315983178674, −5.22811328292776133268304689850, −3.467655211383033621913009524083, −2.47485846053544812411600900965, −1.491569146846128940265810489972, 0.40455151700596809180418675790, 2.53420502650915456791347833638, 3.12499357621244699405735243725, 4.78384164068328018957849250526, 5.30774441416288864864504275738, 6.17421762550209573771541204614, 7.49890599393077972610418144721, 8.92179522914960765357812016143, 9.46818702367328195512796106046, 10.1838545822220329233405677518, 11.10426461226079165297263586509, 12.34970058809382389008523281328, 13.0798924345315056621441122117, 14.01781741942647275774960692066, 15.15524664148697158189359762350, 15.90366229812139315266705284806, 16.53025671815887685105245677819, 17.47877842032312137720229063456, 18.223323859831681542352814781333, 19.40529852376643169204902632288, 20.51529670405613850859520941415, 20.8579202193201656043309424193, 22.03051356714803080589028618203, 22.34696337338550492353929177609, 23.304457772158165473574400949495

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