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Properties

Label 1-1441-1441.1076-r0-0-0
Degree $1$
Conductor $1441$
Sign $-0.694 + 0.719i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.607 + 0.794i)2-s + (0.885 + 0.464i)3-s + (−0.262 − 0.964i)4-s + (0.885 + 0.464i)5-s + (−0.906 + 0.421i)6-s + (−0.906 − 0.421i)7-s + (0.926 + 0.377i)8-s + (0.568 + 0.822i)9-s + (−0.906 + 0.421i)10-s + (0.215 − 0.976i)12-s + (0.485 + 0.873i)13-s + (0.885 − 0.464i)14-s + (0.568 + 0.822i)15-s + (−0.861 + 0.506i)16-s + (0.399 − 0.916i)17-s + (−0.998 − 0.0483i)18-s + ⋯
L(s)  = 1  + (−0.607 + 0.794i)2-s + (0.885 + 0.464i)3-s + (−0.262 − 0.964i)4-s + (0.885 + 0.464i)5-s + (−0.906 + 0.421i)6-s + (−0.906 − 0.421i)7-s + (0.926 + 0.377i)8-s + (0.568 + 0.822i)9-s + (−0.906 + 0.421i)10-s + (0.215 − 0.976i)12-s + (0.485 + 0.873i)13-s + (0.885 − 0.464i)14-s + (0.568 + 0.822i)15-s + (−0.861 + 0.506i)16-s + (0.399 − 0.916i)17-s + (−0.998 − 0.0483i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.694 + 0.719i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.694 + 0.719i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $-0.694 + 0.719i$
Analytic conductor: \(6.69197\)
Root analytic conductor: \(6.69197\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (1076, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (0:\ ),\ -0.694 + 0.719i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6319310796 + 1.489324237i\)
\(L(\frac12)\) \(\approx\) \(0.6319310796 + 1.489324237i\)
\(L(1)\) \(\approx\) \(0.9060511631 + 0.6874795371i\)
\(L(1)\) \(\approx\) \(0.9060511631 + 0.6874795371i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (-0.607 + 0.794i)T \)
3 \( 1 + (0.885 + 0.464i)T \)
5 \( 1 + (0.885 + 0.464i)T \)
7 \( 1 + (-0.906 - 0.421i)T \)
13 \( 1 + (0.485 + 0.873i)T \)
17 \( 1 + (0.399 - 0.916i)T \)
19 \( 1 + (0.399 + 0.916i)T \)
23 \( 1 + (-0.527 + 0.849i)T \)
29 \( 1 + (-0.607 - 0.794i)T \)
31 \( 1 + (-0.970 + 0.239i)T \)
37 \( 1 + (0.779 + 0.626i)T \)
41 \( 1 + (0.215 - 0.976i)T \)
43 \( 1 + (-0.989 + 0.144i)T \)
47 \( 1 + (0.958 - 0.285i)T \)
53 \( 1 + (0.309 + 0.951i)T \)
59 \( 1 + (0.995 + 0.0965i)T \)
61 \( 1 + (0.309 + 0.951i)T \)
67 \( 1 + (-0.989 - 0.144i)T \)
71 \( 1 + (0.644 - 0.764i)T \)
73 \( 1 + (-0.809 + 0.587i)T \)
79 \( 1 + (-0.443 - 0.896i)T \)
83 \( 1 + (-0.354 - 0.935i)T \)
89 \( 1 + (-0.809 + 0.587i)T \)
97 \( 1 + (0.120 - 0.992i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−20.17906381997108447198753370965, −19.90015146317804112912120964142, −18.973686246887610193393694033901, −18.20636357105219431863249891625, −17.87634470814930717012945104274, −16.76196754435924283470416736838, −16.12892480804193150187308123311, −15.094315670560036140410788636641, −14.14190909333999266039629829855, −13.12908121064036761414793848332, −12.93152640897153639440194232795, −12.34997902150633652047034788811, −11.11248209970667120606931359900, −10.0853452893463276640643433032, −9.64476803982647777828535581039, −8.79290438894877312529354079171, −8.37978156726478648143791581718, −7.32203461046578731256343224273, −6.394431585966814726199284549337, −5.46007390206472752681664074170, −4.07157458689834621395503213173, −3.216071049386280859904769254206, −2.51163837948268883166678265889, −1.68864699558628645573420715116, −0.68928731463570225920241556990, 1.34210462812283551833318613058, 2.24352847937930980629188382237, 3.371936953522638096282091196792, 4.20416012288891735786087836474, 5.45979129825549813331272220321, 6.10087168214301280917753082152, 7.15915936535475316179531566119, 7.55153661557494146451411343006, 8.81079522699815335208769683220, 9.39202045674116661324372197618, 9.94522352517674525681030960900, 10.48081316630716692620290227155, 11.5965502772072602532488460646, 13.15799130342492447707085476709, 13.727479153231941955088227171027, 14.1657107753219715667082897452, 14.96543237276064998865147426775, 15.84784498059787610526681356231, 16.46080791318709516415810713181, 16.975966393351209027959422942445, 18.19390075120279687301280606089, 18.67645539257288460873486586071, 19.30467569839134823609365948002, 20.22602470097265923792149178020, 20.81034568349567425687728386109

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