↑

Properties

Label 1-1480-1480.27-r0-0-0
Degree $1$
Conductor $1480$
Sign $0.261 + 0.965i$
Analytic cond. $6.87309$
Root an. cond. $6.87309$
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  + (0.866 + 0.5i)3-s + (0.866 + 0.5i)7-s + (0.5 + 0.866i)9-s + 11-s + (0.866 + 0.5i)13-s + (−0.866 + 0.5i)17-s + (−0.5 + 0.866i)19-s + (0.5 + 0.866i)21-s − i·23-s + i·27-s − 29-s + 31-s + (0.866 + 0.5i)33-s + (0.5 + 0.866i)39-s + (−0.5 + 0.866i)41-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)3-s + (0.866 + 0.5i)7-s + (0.5 + 0.866i)9-s + 11-s + (0.866 + 0.5i)13-s + (−0.866 + 0.5i)17-s + (−0.5 + 0.866i)19-s + (0.5 + 0.866i)21-s − i·23-s + i·27-s − 29-s + 31-s + (0.866 + 0.5i)33-s + (0.5 + 0.866i)39-s + (−0.5 + 0.866i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.261 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.261 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1480\)    =    \(2^{3} \cdot 5 \cdot 37\)
Sign: $0.261 + 0.965i$
Analytic conductor: \(6.87309\)
Root analytic conductor: \(6.87309\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1480} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1480,\ (0:\ ),\ 0.261 + 0.965i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.074296294 + 1.587664196i\)
\(L(\frac12)\) \(\approx\) \(2.074296294 + 1.587664196i\)
\(L(1)\) \(\approx\) \(1.560140597 + 0.5479653961i\)
\(L(1)\) \(\approx\) \(1.560140597 + 0.5479653961i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
37 \( 1 \)
good3 \( 1 + (0.866 + 0.5i)T \)
7 \( 1 + (0.866 + 0.5i)T \)
11 \( 1 + T \)
13 \( 1 + (0.866 + 0.5i)T \)
17 \( 1 + (-0.866 + 0.5i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 - iT \)
29 \( 1 - T \)
31 \( 1 + T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + iT \)
47 \( 1 - iT \)
53 \( 1 + (0.866 - 0.5i)T \)
59 \( 1 + (-0.5 - 0.866i)T \)
61 \( 1 + (-0.5 + 0.866i)T \)
67 \( 1 + (-0.866 - 0.5i)T \)
71 \( 1 + (0.5 - 0.866i)T \)
73 \( 1 - iT \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + (-0.866 + 0.5i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 - iT \)
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

−20.35712226953831912957126569246, −19.88528968313039133205469653382, −19.09572268238667051285096383458, −18.2594088248092143712669228724, −17.542814267654468884524480496922, −17.043297322213512687175339469720, −15.63494872942117979512582320520, −15.249454889782546505527708910107, −14.31265970416165260448396718637, −13.63886035027610008288997569379, −13.23584745963511944762055479798, −12.08849628761737763881307615611, −11.35775652476355746904280410954, −10.61819639646472963801461940067, −9.42765608658270902722637008093, −8.83133226979306875041186710678, −8.128343512414042149428081792043, −7.22775767544435726145915795240, −6.668175719722171672557294790204, −5.55670034398998495665965233523, −4.328915023182830113214482052459, −3.78452080064938332153321202529, −2.6832275917916785062706246448, −1.70233694643334073589868772920, −0.928040556886346026731283338981, 1.53723684208592308492186726529, 2.04649404605046382904372779818, 3.25025136844035301372177342287, 4.18963664040104006750495487647, 4.62912647028046475037575250980, 5.93379734827555269071192284832, 6.68493586396909130467599986321, 7.89615479460669067712112548663, 8.57798413383891908164535484943, 8.97606320414657687759876005266, 9.97401900096236369855499952070, 10.87082745597060608256055351721, 11.51509250718303814107960034053, 12.462797183757050814377598647118, 13.448159281307343687415458019379, 14.10444173470299571841473832445, 14.94784111228494836488681458056, 15.16048079650482175599708826689, 16.38005742354501361404149249105, 16.83967363913511006444244649505, 17.976891963632462247230056142475, 18.641821304021582548661475825540, 19.389300538161549025680249555382, 20.13416915356659572671615522752, 20.91843716954783717179260106537

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