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Properties

Label 1-1441-1441.1064-r1-0-0
Degree $1$
Conductor $1441$
Sign $0.326 - 0.945i$
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  + (−0.485 + 0.873i)2-s + (−0.748 − 0.663i)3-s + (−0.527 − 0.849i)4-s + (−0.748 − 0.663i)5-s + (0.943 − 0.331i)6-s + (0.943 + 0.331i)7-s + (0.998 − 0.0483i)8-s + (0.120 + 0.992i)9-s + (0.943 − 0.331i)10-s + (−0.168 + 0.985i)12-s + (0.607 − 0.794i)13-s + (−0.748 + 0.663i)14-s + (0.120 + 0.992i)15-s + (−0.443 + 0.896i)16-s + (0.989 + 0.144i)17-s + (−0.926 − 0.377i)18-s + ⋯
L(s)  = 1  + (−0.485 + 0.873i)2-s + (−0.748 − 0.663i)3-s + (−0.527 − 0.849i)4-s + (−0.748 − 0.663i)5-s + (0.943 − 0.331i)6-s + (0.943 + 0.331i)7-s + (0.998 − 0.0483i)8-s + (0.120 + 0.992i)9-s + (0.943 − 0.331i)10-s + (−0.168 + 0.985i)12-s + (0.607 − 0.794i)13-s + (−0.748 + 0.663i)14-s + (0.120 + 0.992i)15-s + (−0.443 + 0.896i)16-s + (0.989 + 0.144i)17-s + (−0.926 − 0.377i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.326 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.326 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $0.326 - 0.945i$
Analytic conductor: \(154.856\)
Root analytic conductor: \(154.856\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (1064, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (1:\ ),\ 0.326 - 0.945i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.019662433 - 0.7263743469i\)
\(L(\frac12)\) \(\approx\) \(1.019662433 - 0.7263743469i\)
\(L(1)\) \(\approx\) \(0.7122415166 - 0.06039902172i\)
\(L(1)\) \(\approx\) \(0.7122415166 - 0.06039902172i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (-0.485 + 0.873i)T \)
3 \( 1 + (-0.748 - 0.663i)T \)
5 \( 1 + (-0.748 - 0.663i)T \)
7 \( 1 + (0.943 + 0.331i)T \)
13 \( 1 + (0.607 - 0.794i)T \)
17 \( 1 + (0.989 + 0.144i)T \)
19 \( 1 + (0.989 - 0.144i)T \)
23 \( 1 + (-0.262 - 0.964i)T \)
29 \( 1 + (-0.485 - 0.873i)T \)
31 \( 1 + (-0.354 - 0.935i)T \)
37 \( 1 + (0.644 - 0.764i)T \)
41 \( 1 + (0.168 - 0.985i)T \)
43 \( 1 + (-0.399 + 0.916i)T \)
47 \( 1 + (-0.681 - 0.732i)T \)
53 \( 1 + (-0.809 - 0.587i)T \)
59 \( 1 + (0.715 + 0.698i)T \)
61 \( 1 + (0.809 + 0.587i)T \)
67 \( 1 + (0.399 + 0.916i)T \)
71 \( 1 + (0.779 - 0.626i)T \)
73 \( 1 + (-0.309 - 0.951i)T \)
79 \( 1 + (0.861 - 0.506i)T \)
83 \( 1 + (0.970 + 0.239i)T \)
89 \( 1 + (0.309 + 0.951i)T \)
97 \( 1 + (0.568 + 0.822i)T \)
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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

−20.567263682669217243910175681335, −20.17412994489370247457456887754, −19.07205616355462023989388015108, −18.319769360283679268320496785939, −17.94905192876304904895971954764, −16.96296239690414926940776891039, −16.32069010114273249878385408979, −15.62311663377272124144802286192, −14.457083490352250928303233643465, −14.022115511589232070656999784865, −12.69666166697977192168567214149, −11.72684755704203896011402558826, −11.47156330170254983268679092052, −10.86178602599598018511298392405, −10.045112306616377801800196428723, −9.33787381510346823596193051721, −8.26860485236812723431066143694, −7.54839456248166908708862388168, −6.71143835247674552883710736650, −5.3483650175209197885457986335, −4.60597022417402355125563900748, −3.64756175911349438558181930265, −3.26202607740634890816373097469, −1.645035139208978988970944395976, −0.87121960028628454361361048971, 0.50863963517588941428151973828, 0.9979214252248632034084208489, 2.081691083177258812614414957802, 3.84193046475307324208627839224, 4.87213213594101103532824676535, 5.462533625512896171731862741262, 6.094182662043645655729762246782, 7.30753255792194704658587673991, 7.9373823989591036431819345735, 8.25062170810431799297351770634, 9.343845351273418633423152119455, 10.412807241541834952297405898105, 11.250945776705043211213315924080, 11.86421970549328500794294879930, 12.80443138377847159200997260192, 13.4953776121484054470551993976, 14.53184817555764056141024131261, 15.201166892166696692863720377160, 16.18210918561772209509418343673, 16.50800794982015481123753700831, 17.45282460963433951496056887064, 18.02844755994271952467394723660, 18.65135048611847710420483065850, 19.33486650097443753000144794577, 20.2812829912199667838736919436

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