↑

Properties

Label 1-483-483.215-r0-0-0
Degree $1$
Conductor $483$
Sign $0.958 + 0.286i$
Analytic cond. $2.24304$
Root an. cond. $2.24304$
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  + (−0.235 − 0.971i)2-s + (−0.888 + 0.458i)4-s + (0.928 − 0.371i)5-s + (0.654 + 0.755i)8-s + (−0.580 − 0.814i)10-s + (−0.235 + 0.971i)11-s + (−0.415 + 0.909i)13-s + (0.580 − 0.814i)16-s + (0.0475 + 0.998i)17-s + (−0.0475 + 0.998i)19-s + (−0.654 + 0.755i)20-s + 22-s + (0.723 − 0.690i)25-s + (0.981 + 0.189i)26-s + (−0.841 + 0.540i)29-s + ⋯
L(s)  = 1  + (−0.235 − 0.971i)2-s + (−0.888 + 0.458i)4-s + (0.928 − 0.371i)5-s + (0.654 + 0.755i)8-s + (−0.580 − 0.814i)10-s + (−0.235 + 0.971i)11-s + (−0.415 + 0.909i)13-s + (0.580 − 0.814i)16-s + (0.0475 + 0.998i)17-s + (−0.0475 + 0.998i)19-s + (−0.654 + 0.755i)20-s + 22-s + (0.723 − 0.690i)25-s + (0.981 + 0.189i)26-s + (−0.841 + 0.540i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.958 + 0.286i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.958 + 0.286i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.958 + 0.286i$
Analytic conductor: \(2.24304\)
Root analytic conductor: \(2.24304\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (215, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 483,\ (0:\ ),\ 0.958 + 0.286i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.026693912 + 0.1503554103i\)
\(L(\frac12)\) \(\approx\) \(1.026693912 + 0.1503554103i\)
\(L(1)\) \(\approx\) \(0.9106118086 - 0.1884941578i\)
\(L(1)\) \(\approx\) \(0.9106118086 - 0.1884941578i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.235 - 0.971i)T \)
5 \( 1 + (0.928 - 0.371i)T \)
11 \( 1 + (-0.235 + 0.971i)T \)
13 \( 1 + (-0.415 + 0.909i)T \)
17 \( 1 + (0.0475 + 0.998i)T \)
19 \( 1 + (-0.0475 + 0.998i)T \)
29 \( 1 + (-0.841 + 0.540i)T \)
31 \( 1 + (-0.981 + 0.189i)T \)
37 \( 1 + (-0.786 + 0.618i)T \)
41 \( 1 + (-0.142 + 0.989i)T \)
43 \( 1 + (-0.654 + 0.755i)T \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 + (0.995 + 0.0950i)T \)
59 \( 1 + (0.580 + 0.814i)T \)
61 \( 1 + (0.327 - 0.945i)T \)
67 \( 1 + (0.723 - 0.690i)T \)
71 \( 1 + (0.959 + 0.281i)T \)
73 \( 1 + (0.888 - 0.458i)T \)
79 \( 1 + (-0.995 + 0.0950i)T \)
83 \( 1 + (-0.142 - 0.989i)T \)
89 \( 1 + (0.981 + 0.189i)T \)
97 \( 1 + (0.142 - 0.989i)T \)
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

−24.06017941194275321599853248171, −22.79810833401676499242551038142, −22.255859084084446866528786599845, −21.43895420029662739612963613014, −20.313670645714277051900461563383, −19.14990124361275433186507262207, −18.398198626656067327731014903160, −17.66266277501993454534543259299, −16.96070774683430392528497516700, −16.01094620675884987013243328601, −15.17008128643235543390954084922, −14.249915648683402470256506038339, −13.54314142617713699730635153840, −12.84564221963674228025279332584, −11.22409387046760152862342482835, −10.33421988667860512861875291924, −9.44668859013192323926981550181, −8.6699916410980997301361978583, −7.50141417009894515863676712547, −6.747884998465700401560755044661, −5.5736001600159675503359964858, −5.210616881564322948127401108355, −3.56941418439796708499496846534, −2.32473442560037577001511804546, −0.61714881807669579866456906235, 1.62282546046921100690563079582, 2.032857849175445027699322249352, 3.51693243461690666747986611688, 4.60193012101101234278196272047, 5.475065265280835623775799139648, 6.80882625104919941062654773684, 8.064667388095371332015215548139, 9.031586721431938759844494248107, 9.84799258619835322105823008759, 10.41438786664134492585261464508, 11.62005478871187813110703200884, 12.57065414664852770636175156798, 13.07018088661368133873738767903, 14.17149717031665430859460051836, 14.87614608557043601101979623510, 16.59556667629615075860470371606, 17.005835279602094070235316915063, 18.034771884902231506972431541380, 18.619287358225623059494236099447, 19.74404470800768953150216010967, 20.42094578442420247325716391909, 21.30205986614083958682039077463, 21.77115339556011593169485349359, 22.7688681222807851146588865073, 23.65857434464802950614249829831

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