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Properties

Label 1-1441-1441.1016-r0-0-0
Degree $1$
Conductor $1441$
Sign $0.520 - 0.853i$
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  + (−0.0724 − 0.997i)2-s + (−0.681 + 0.732i)3-s + (−0.989 + 0.144i)4-s + (−0.906 − 0.421i)5-s + (0.779 + 0.626i)6-s + (0.836 + 0.548i)7-s + (0.215 + 0.976i)8-s + (−0.0724 − 0.997i)9-s + (−0.354 + 0.935i)10-s + (0.568 − 0.822i)12-s + (0.779 − 0.626i)13-s + (0.485 − 0.873i)14-s + (0.926 − 0.377i)15-s + (0.958 − 0.285i)16-s + (0.958 − 0.285i)17-s + (−0.989 + 0.144i)18-s + ⋯
L(s)  = 1  + (−0.0724 − 0.997i)2-s + (−0.681 + 0.732i)3-s + (−0.989 + 0.144i)4-s + (−0.906 − 0.421i)5-s + (0.779 + 0.626i)6-s + (0.836 + 0.548i)7-s + (0.215 + 0.976i)8-s + (−0.0724 − 0.997i)9-s + (−0.354 + 0.935i)10-s + (0.568 − 0.822i)12-s + (0.779 − 0.626i)13-s + (0.485 − 0.873i)14-s + (0.926 − 0.377i)15-s + (0.958 − 0.285i)16-s + (0.958 − 0.285i)17-s + (−0.989 + 0.144i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.520 - 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.520 - 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $0.520 - 0.853i$
Analytic conductor: \(6.69197\)
Root analytic conductor: \(6.69197\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (1016, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (0:\ ),\ 0.520 - 0.853i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7653209611 - 0.4297386431i\)
\(L(\frac12)\) \(\approx\) \(0.7653209611 - 0.4297386431i\)
\(L(1)\) \(\approx\) \(0.6896638293 - 0.2253304801i\)
\(L(1)\) \(\approx\) \(0.6896638293 - 0.2253304801i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (-0.0724 - 0.997i)T \)
3 \( 1 + (-0.681 + 0.732i)T \)
5 \( 1 + (-0.906 - 0.421i)T \)
7 \( 1 + (0.836 + 0.548i)T \)
13 \( 1 + (0.779 - 0.626i)T \)
17 \( 1 + (0.958 - 0.285i)T \)
19 \( 1 + (-0.943 + 0.331i)T \)
23 \( 1 + (-0.748 + 0.663i)T \)
29 \( 1 + (0.644 - 0.764i)T \)
31 \( 1 + (0.399 - 0.916i)T \)
37 \( 1 + (-0.989 + 0.144i)T \)
41 \( 1 + (-0.943 + 0.331i)T \)
43 \( 1 + (0.120 - 0.992i)T \)
47 \( 1 + (0.644 + 0.764i)T \)
53 \( 1 + (0.309 + 0.951i)T \)
59 \( 1 + (-0.943 - 0.331i)T \)
61 \( 1 + (0.309 - 0.951i)T \)
67 \( 1 + (0.120 + 0.992i)T \)
71 \( 1 + (-0.861 + 0.506i)T \)
73 \( 1 + (-0.809 + 0.587i)T \)
79 \( 1 + (0.981 - 0.192i)T \)
83 \( 1 + (-0.168 - 0.985i)T \)
89 \( 1 + T \)
97 \( 1 + (0.779 - 0.626i)T \)
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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

−21.00128841279691574517474658884, −19.68275126422774030035383943830, −19.16407949480873175203993126064, −18.30682133044443998117585859482, −17.94244504208233233057337383503, −16.95890885900104466863565729881, −16.451078395855561608126892632074, −15.71312542588335823535800067487, −14.71052451075416956476620155094, −14.14346057697635098818626418505, −13.44535223971062811948403998629, −12.35757151955010015095892548700, −11.83247692099240923892924764820, −10.66822513620670473905808430967, −10.41516445751539017820920172081, −8.62289047048252572369403998981, −8.27899755406312758017488410722, −7.41013638026778896418412850543, −6.79307100413981993829694680313, −6.149780374085478799359269136, −5.02027306908410217226283565427, −4.38413537521531574578182616582, −3.4475331395136345989268659468, −1.76152172834836014723299969782, −0.75536844295958159333183809729, 0.621083792063054587508344155482, 1.63517756483467471509683074956, 3.013963650193695542683853388734, 3.868858635910709644363130962555, 4.48014316187428920709034825024, 5.33026202068075642844055869801, 5.99133103994772107240328114486, 7.69221651530063876878579613246, 8.35828215128627930969294748794, 9.01220366983421818573858248718, 10.05839413615200276878251733670, 10.68136373164935127074692297886, 11.5127188057462553873530805112, 11.95641679420574156293463076149, 12.52051484797567779022170814112, 13.6315275490963423769144140895, 14.62938227320619584818658545225, 15.38713544389841028732849091352, 15.99867372330814690045916488878, 17.15996490130489456380439386745, 17.45341475834057657162542420588, 18.63548565683470265558870030312, 18.91111989204496006688293424683, 20.20494557194830265197711699884, 20.629962822878707993063391582147

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