↑

Properties

Label 1-1441-1441.17-r0-0-0
Degree $1$
Conductor $1441$
Sign $0.468 + 0.883i$
Analytic cond. $6.69197$
Root an. cond. $6.69197$
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  + (0.120 + 0.992i)2-s + (0.995 + 0.0965i)3-s + (−0.970 + 0.239i)4-s + (0.399 − 0.916i)5-s + (0.0241 + 0.999i)6-s + (0.943 + 0.331i)7-s + (−0.354 − 0.935i)8-s + (0.981 + 0.192i)9-s + (0.958 + 0.285i)10-s + (−0.989 + 0.144i)12-s + (0.607 − 0.794i)13-s + (−0.215 + 0.976i)14-s + (0.485 − 0.873i)15-s + (0.885 − 0.464i)16-s + (−0.443 + 0.896i)17-s + (−0.0724 + 0.997i)18-s + ⋯
L(s)  = 1  + (0.120 + 0.992i)2-s + (0.995 + 0.0965i)3-s + (−0.970 + 0.239i)4-s + (0.399 − 0.916i)5-s + (0.0241 + 0.999i)6-s + (0.943 + 0.331i)7-s + (−0.354 − 0.935i)8-s + (0.981 + 0.192i)9-s + (0.958 + 0.285i)10-s + (−0.989 + 0.144i)12-s + (0.607 − 0.794i)13-s + (−0.215 + 0.976i)14-s + (0.485 − 0.873i)15-s + (0.885 − 0.464i)16-s + (−0.443 + 0.896i)17-s + (−0.0724 + 0.997i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.468 + 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.468 + 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $0.468 + 0.883i$
Analytic conductor: \(6.69197\)
Root analytic conductor: \(6.69197\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (0:\ ),\ 0.468 + 0.883i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.359186741 + 1.419979284i\)
\(L(\frac12)\) \(\approx\) \(2.359186741 + 1.419979284i\)
\(L(1)\) \(\approx\) \(1.582538237 + 0.7022675948i\)
\(L(1)\) \(\approx\) \(1.582538237 + 0.7022675948i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (0.120 + 0.992i)T \)
3 \( 1 + (0.995 + 0.0965i)T \)
5 \( 1 + (0.399 - 0.916i)T \)
7 \( 1 + (0.943 + 0.331i)T \)
13 \( 1 + (0.607 - 0.794i)T \)
17 \( 1 + (-0.443 + 0.896i)T \)
19 \( 1 + (-0.168 + 0.985i)T \)
23 \( 1 + (-0.779 - 0.626i)T \)
29 \( 1 + (0.485 + 0.873i)T \)
31 \( 1 + (0.998 - 0.0483i)T \)
37 \( 1 + (-0.644 + 0.764i)T \)
41 \( 1 + (-0.715 + 0.698i)T \)
43 \( 1 + (-0.995 - 0.0965i)T \)
47 \( 1 + (0.906 - 0.421i)T \)
53 \( 1 + T \)
59 \( 1 + (0.885 - 0.464i)T \)
61 \( 1 + (0.809 - 0.587i)T \)
67 \( 1 + (-0.995 + 0.0965i)T \)
71 \( 1 + (0.262 - 0.964i)T \)
73 \( 1 + T \)
79 \( 1 + (0.995 + 0.0965i)T \)
83 \( 1 + (-0.0724 - 0.997i)T \)
89 \( 1 + (-0.809 + 0.587i)T \)
97 \( 1 + (-0.958 + 0.285i)T \)
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

−20.7864756351333301801801817401, −19.78098904539513061545406628570, −19.280546218122453775111058810786, −18.39786616467824093774296622166, −17.96835112784274268858012685927, −17.2468012567834274122119835549, −15.72686557361711053114773201791, −15.04855477697679456507934888021, −14.13449824824789163705700003502, −13.74222028837020973899296115658, −13.38091951892464219511790859282, −11.93432551747767399757977867561, −11.4255999264468586430004756709, −10.54985044115467396939756994113, −9.88623926390691227684815523917, −9.04001276374669963262553562654, −8.378801930751287195121726533206, −7.38050898274361117738766653481, −6.58452783732289073308527886758, −5.28027527425359373575728747632, −4.27587577334076472432924438723, −3.68481531740533509834706081277, −2.523499695102220981494934614955, −2.15021966671199661900295226563, −1.102643008306919706909566084697, 1.163984469934560537252430894347, 2.03600589177158208180179864400, 3.4131148429142096615189756137, 4.27962622090047325706609697305, 4.97848121816928896236020206072, 5.819566145392608433250278909744, 6.71601148668503629252950878202, 8.02164929478988523550058381956, 8.38401634962520912975033577621, 8.67976553238719884048929272758, 9.88525466332205841383923695751, 10.451803643132851880603615094613, 12.09357109750914619499572885496, 12.683615048523082051326885800404, 13.55525208256672097692301543361, 14.00588080365803680449543658002, 14.95756718721313402826165965164, 15.363858834527087730006268952938, 16.25881458157348384869774136628, 16.956343499429241532747126051709, 17.89717474253507061458042508235, 18.315050617195082875964256376288, 19.31655694904506815364727137061, 20.333844967578159160219661998, 20.81565094071596973250742772653

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