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Properties

Label 1-1441-1441.1021-r1-0-0
Degree $1$
Conductor $1441$
Sign $0.452 - 0.891i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.0724 − 0.997i)2-s + (−0.906 + 0.421i)3-s + (−0.989 − 0.144i)4-s + (0.485 − 0.873i)5-s + (0.354 + 0.935i)6-s + (−0.262 − 0.964i)7-s + (−0.215 + 0.976i)8-s + (0.644 − 0.764i)9-s + (−0.836 − 0.548i)10-s + (0.958 − 0.285i)12-s + (−0.998 − 0.0483i)13-s + (−0.981 + 0.192i)14-s + (−0.0724 + 0.997i)15-s + (0.958 + 0.285i)16-s + (−0.568 + 0.822i)17-s + (−0.715 − 0.698i)18-s + ⋯
L(s)  = 1  + (0.0724 − 0.997i)2-s + (−0.906 + 0.421i)3-s + (−0.989 − 0.144i)4-s + (0.485 − 0.873i)5-s + (0.354 + 0.935i)6-s + (−0.262 − 0.964i)7-s + (−0.215 + 0.976i)8-s + (0.644 − 0.764i)9-s + (−0.836 − 0.548i)10-s + (0.958 − 0.285i)12-s + (−0.998 − 0.0483i)13-s + (−0.981 + 0.192i)14-s + (−0.0724 + 0.997i)15-s + (0.958 + 0.285i)16-s + (−0.568 + 0.822i)17-s + (−0.715 − 0.698i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.452 - 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.452 - 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $0.452 - 0.891i$
Analytic conductor: \(154.856\)
Root analytic conductor: \(154.856\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (1021, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (1:\ ),\ 0.452 - 0.891i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5334662468 - 0.3275597548i\)
\(L(\frac12)\) \(\approx\) \(0.5334662468 - 0.3275597548i\)
\(L(1)\) \(\approx\) \(0.5137686440 - 0.3667319481i\)
\(L(1)\) \(\approx\) \(0.5137686440 - 0.3667319481i\)

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 \( 1 + (-0.906 + 0.421i)T \)
5 \( 1 + (0.485 - 0.873i)T \)
7 \( 1 + (-0.262 - 0.964i)T \)
13 \( 1 + (-0.998 - 0.0483i)T \)
17 \( 1 + (-0.568 + 0.822i)T \)
19 \( 1 + (-0.958 + 0.285i)T \)
23 \( 1 + (-0.215 - 0.976i)T \)
29 \( 1 + (-0.926 + 0.377i)T \)
31 \( 1 + (-0.215 + 0.976i)T \)
37 \( 1 + (0.443 - 0.896i)T \)
41 \( 1 + (0.568 + 0.822i)T \)
43 \( 1 + (0.981 + 0.192i)T \)
47 \( 1 + (0.0724 - 0.997i)T \)
53 \( 1 + (0.309 - 0.951i)T \)
59 \( 1 + (-0.943 + 0.331i)T \)
61 \( 1 + T \)
67 \( 1 + (-0.981 + 0.192i)T \)
71 \( 1 + (-0.215 + 0.976i)T \)
73 \( 1 + (0.809 + 0.587i)T \)
79 \( 1 + (-0.485 + 0.873i)T \)
83 \( 1 + (0.443 + 0.896i)T \)
89 \( 1 + (0.309 - 0.951i)T \)
97 \( 1 + (0.262 + 0.964i)T \)
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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

−21.06733022294179072749291960445, −19.38972016540251571945229256516, −18.88134111733173349421755296367, −18.27289364736032612357997653002, −17.50535403009609199149166508905, −17.12860452239078628133685181816, −16.12759069690452416870613791316, −15.3379130877332862443933261045, −14.85234721639538807851815543853, −13.79206439793001828382898261334, −13.1808391723590008633316744642, −12.36634742348909589806811687074, −11.56240642067071465172708775696, −10.66964204986881645216463515271, −9.60099883268829557161021965215, −9.19827436574319928741495182394, −7.74485598386277325686120954625, −7.2701656929944741624188236821, −6.34534109709058188109204155767, −5.90107341300013431903495993662, −5.132704093319783707049565373130, −4.213482449667424479496967325269, −2.788604358993471379934460140638, −1.9519803622029674228204213626, −0.27212119274109408042926839606, 0.46847272303703852085236734793, 1.408693784405684753450675561702, 2.391873045712712141220675416374, 3.89491332697289094071452058583, 4.29728808111241708093385673255, 5.105790723712306499132638750462, 5.92726497404022899292207926225, 6.89203559577428343064555514110, 8.17094656066826949817682324336, 9.11345339223776174722075806962, 9.81232143540286025770735194173, 10.49960341595017736185985093437, 10.97190833760640677542355651934, 12.08115112604718024293400058316, 12.80754299900205273365624128465, 13.00797888392803511084274890483, 14.26123793189029988742100897186, 14.90883911566030452569155153860, 16.2893038798632073062155123684, 16.77152163947818269415594326401, 17.44983316046466653306954062433, 17.926486557744919411908846679380, 19.07410781997102352946611532621, 19.9115759141159812404200045397, 20.3666338661973173982615588619

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