↑

Properties

Label 1-1441-1441.1052-r1-0-0
Degree $1$
Conductor $1441$
Sign $0.926 + 0.376i$
Analytic cond. $154.856$
Root an. cond. $154.856$
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  + (−0.215 − 0.976i)2-s + (−0.262 − 0.964i)3-s + (−0.906 + 0.421i)4-s + (−0.998 − 0.0483i)5-s + (−0.885 + 0.464i)6-s + (−0.715 + 0.698i)7-s + (0.607 + 0.794i)8-s + (−0.861 + 0.506i)9-s + (0.168 + 0.985i)10-s + (0.644 + 0.764i)12-s + (0.989 − 0.144i)13-s + (0.836 + 0.548i)14-s + (0.215 + 0.976i)15-s + (0.644 − 0.764i)16-s + (0.970 − 0.239i)17-s + (0.681 + 0.732i)18-s + ⋯
L(s)  = 1  + (−0.215 − 0.976i)2-s + (−0.262 − 0.964i)3-s + (−0.906 + 0.421i)4-s + (−0.998 − 0.0483i)5-s + (−0.885 + 0.464i)6-s + (−0.715 + 0.698i)7-s + (0.607 + 0.794i)8-s + (−0.861 + 0.506i)9-s + (0.168 + 0.985i)10-s + (0.644 + 0.764i)12-s + (0.989 − 0.144i)13-s + (0.836 + 0.548i)14-s + (0.215 + 0.976i)15-s + (0.644 − 0.764i)16-s + (0.970 − 0.239i)17-s + (0.681 + 0.732i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.926 + 0.376i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.926 + 0.376i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $0.926 + 0.376i$
Analytic conductor: \(154.856\)
Root analytic conductor: \(154.856\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (1052, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (1:\ ),\ 0.926 + 0.376i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1979641307 + 0.03874511723i\)
\(L(\frac12)\) \(\approx\) \(0.1979641307 + 0.03874511723i\)
\(L(1)\) \(\approx\) \(0.4326169756 - 0.3569816651i\)
\(L(1)\) \(\approx\) \(0.4326169756 - 0.3569816651i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (-0.215 - 0.976i)T \)
3 \( 1 + (-0.262 - 0.964i)T \)
5 \( 1 + (-0.998 - 0.0483i)T \)
7 \( 1 + (-0.715 + 0.698i)T \)
13 \( 1 + (0.989 - 0.144i)T \)
17 \( 1 + (0.970 - 0.239i)T \)
19 \( 1 + (-0.644 - 0.764i)T \)
23 \( 1 + (-0.607 + 0.794i)T \)
29 \( 1 + (-0.399 - 0.916i)T \)
31 \( 1 + (-0.607 - 0.794i)T \)
37 \( 1 + (0.981 + 0.192i)T \)
41 \( 1 + (0.970 + 0.239i)T \)
43 \( 1 + (-0.836 + 0.548i)T \)
47 \( 1 + (0.215 + 0.976i)T \)
53 \( 1 + (-0.809 - 0.587i)T \)
59 \( 1 + (-0.527 - 0.849i)T \)
61 \( 1 - T \)
67 \( 1 + (0.836 + 0.548i)T \)
71 \( 1 + (-0.607 - 0.794i)T \)
73 \( 1 + (-0.309 - 0.951i)T \)
79 \( 1 + (0.998 + 0.0483i)T \)
83 \( 1 + (-0.981 + 0.192i)T \)
89 \( 1 + (-0.809 - 0.587i)T \)
97 \( 1 + (0.715 - 0.698i)T \)
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

−20.31535338725737051394479096937, −19.808766103704218020746379985276, −18.79505256712966567321863539506, −18.27177247504142116076080644541, −17.032533016557554745905484137312, −16.44689065424599714131601759515, −16.19736488176663840783597079782, −15.38096315226363529425686830271, −14.59104284040336909706965871037, −14.06744769861267234167397335671, −12.86193535882187432420708476327, −12.165122454105942443477839034827, −10.81900603052169383332682875452, −10.5295091050720794711788668331, −9.586271687292097435254295923966, −8.71258371358338868169068341202, −8.07995367354680472903313160395, −7.13224782304717295488831074960, −6.28573509045695771945177475879, −5.58099524316959632992922869525, −4.431673314941483111347534806108, −3.87352383419300110163011843487, −3.29102698987670052006665074190, −1.11275152403819067481007528746, −0.0777332595259608773771730754, 0.66483485669007840706815938828, 1.71820170287973574430860876917, 2.79098047508693385463833876415, 3.42622915947807002346238227409, 4.4492635396041680761974991702, 5.615239787354456076877553738546, 6.355921830293574939378795657477, 7.6497986913898732823930492001, 8.029498701946792227443673790173, 8.96898955691798872189090982217, 9.71544627448014805966433340635, 11.00166055755972098670501163062, 11.419122662718282948816057375762, 12.104709181484301208931108346566, 12.86876240760614196230759491499, 13.25034641188221640476208797280, 14.29933144332677352470045250475, 15.30554956963601252303867758876, 16.23886494856566914152438516172, 16.93491190713453771754215674230, 17.971578473474646668336422608029, 18.54381948781130906790292811783, 19.11957481788204361477244599264, 19.61503264504731716948759752463, 20.34570771578187929607848802302

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