Properties

Label 1-1441-1441.465-r1-0-0
Degree $1$
Conductor $1441$
Sign $0.708 - 0.706i$
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  + (0.906 − 0.421i)2-s + (0.399 − 0.916i)3-s + (0.644 − 0.764i)4-s + (0.215 + 0.976i)5-s + (−0.0241 − 0.999i)6-s + (0.568 + 0.822i)7-s + (0.262 − 0.964i)8-s + (−0.681 − 0.732i)9-s + (0.607 + 0.794i)10-s + (−0.443 − 0.896i)12-s + (0.568 + 0.822i)13-s + (0.861 + 0.506i)14-s + (0.981 + 0.192i)15-s + (−0.168 − 0.985i)16-s + (0.443 − 0.896i)17-s + (−0.926 − 0.377i)18-s + ⋯
L(s)  = 1  + (0.906 − 0.421i)2-s + (0.399 − 0.916i)3-s + (0.644 − 0.764i)4-s + (0.215 + 0.976i)5-s + (−0.0241 − 0.999i)6-s + (0.568 + 0.822i)7-s + (0.262 − 0.964i)8-s + (−0.681 − 0.732i)9-s + (0.607 + 0.794i)10-s + (−0.443 − 0.896i)12-s + (0.568 + 0.822i)13-s + (0.861 + 0.506i)14-s + (0.981 + 0.192i)15-s + (−0.168 − 0.985i)16-s + (0.443 − 0.896i)17-s + (−0.926 − 0.377i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.708 - 0.706i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.708 - 0.706i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $0.708 - 0.706i$
Analytic conductor: \(154.856\)
Root analytic conductor: \(154.856\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (465, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (1:\ ),\ 0.708 - 0.706i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(5.688428107 - 2.351133463i\)
\(L(\frac12)\) \(\approx\) \(5.688428107 - 2.351133463i\)
\(L(1)\) \(\approx\) \(2.362881411 - 0.8371206817i\)
\(L(1)\) \(\approx\) \(2.362881411 - 0.8371206817i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (-0.906 + 0.421i)T \)
3 \( 1 + (-0.399 + 0.916i)T \)
5 \( 1 + (-0.215 - 0.976i)T \)
7 \( 1 + (-0.568 - 0.822i)T \)
13 \( 1 + (-0.568 - 0.822i)T \)
17 \( 1 + (-0.443 + 0.896i)T \)
19 \( 1 + (-0.989 + 0.144i)T \)
23 \( 1 + (-0.998 - 0.0483i)T \)
29 \( 1 + (0.120 - 0.992i)T \)
31 \( 1 + (0.836 + 0.548i)T \)
37 \( 1 + (-0.970 + 0.239i)T \)
41 \( 1 + (-0.715 + 0.698i)T \)
43 \( 1 + (-0.215 - 0.976i)T \)
47 \( 1 + (-0.681 - 0.732i)T \)
53 \( 1 + (-0.309 - 0.951i)T \)
59 \( 1 + (0.443 - 0.896i)T \)
61 \( 1 + (0.809 - 0.587i)T \)
67 \( 1 + (0.215 - 0.976i)T \)
71 \( 1 + (0.779 - 0.626i)T \)
73 \( 1 + (-0.809 + 0.587i)T \)
79 \( 1 + (-0.861 + 0.506i)T \)
83 \( 1 + (0.644 + 0.764i)T \)
89 \( 1 + (0.809 + 0.587i)T \)
97 \( 1 + (-0.943 - 0.331i)T \)
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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

−20.97697735342156788158196635602, −20.10700951973484701003483749675, −19.7408384216288327728724067891, −18.119735011018896641926721011439, −16.99628249851102514552229774654, −16.92270123661646101773677620661, −15.976266385251934632465084047257, −15.31523500902711686559994012925, −14.58311443367637762723888482157, −13.79606860978174493257382350980, −13.25582099625217810544429453406, −12.45561923578790663997914599191, −11.37766930868410626360882339255, −10.744199857070253052136135117305, −9.84191691283522781455957292628, −8.81446588427600905494922718617, −8.04901917517613219476882580892, −7.55547691433727518966602855257, −6.10908170011023942107807936606, −5.326728837968198861725811260708, −4.78534348507781348503672506110, −3.87624740858219692440355176488, −3.347796625357790652180843940630, −2.04581515996316259015412831822, −0.892064489162832159427923228938, 1.03057451921507650010654959702, 1.83483152417821968513059010205, 2.76445715651086270376549456663, 3.15960997458762750539087812473, 4.41500963521746310077390704050, 5.64533383557581685544401977335, 6.026282897106735365400714359857, 7.23807107996351379895014563875, 7.42083478153459217610582107910, 8.989442382810807986548018249711, 9.48905716423894732796202713825, 10.89329800102144209788288516533, 11.39059980780680708711068500588, 12.00272220216500682872450977121, 12.87477345303078255742418290284, 13.71443283856169651593476663179, 14.2773927258361496169001379649, 14.75468609008538058349723039172, 15.5635567751483982196683572626, 16.53643812421850751468686790036, 17.91546052563765823080306085702, 18.42543809452934315314520617558, 18.85977390969553091407750707299, 19.672765982547720309722480816798, 20.60455326485619178874678612805

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