Properties

Label 1-1441-1441.1081-r0-0-0
Degree $1$
Conductor $1441$
Sign $0.986 - 0.164i$
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  + (0.215 − 0.976i)2-s + (−0.262 + 0.964i)3-s + (−0.906 − 0.421i)4-s + (−0.998 + 0.0483i)5-s + (0.885 + 0.464i)6-s + (0.715 + 0.698i)7-s + (−0.607 + 0.794i)8-s + (−0.861 − 0.506i)9-s + (−0.168 + 0.985i)10-s + (0.644 − 0.764i)12-s + (−0.989 − 0.144i)13-s + (0.836 − 0.548i)14-s + (0.215 − 0.976i)15-s + (0.644 + 0.764i)16-s + (−0.970 − 0.239i)17-s + (−0.681 + 0.732i)18-s + ⋯
L(s)  = 1  + (0.215 − 0.976i)2-s + (−0.262 + 0.964i)3-s + (−0.906 − 0.421i)4-s + (−0.998 + 0.0483i)5-s + (0.885 + 0.464i)6-s + (0.715 + 0.698i)7-s + (−0.607 + 0.794i)8-s + (−0.861 − 0.506i)9-s + (−0.168 + 0.985i)10-s + (0.644 − 0.764i)12-s + (−0.989 − 0.144i)13-s + (0.836 − 0.548i)14-s + (0.215 − 0.976i)15-s + (0.644 + 0.764i)16-s + (−0.970 − 0.239i)17-s + (−0.681 + 0.732i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.164i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.164i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $0.986 - 0.164i$
Analytic conductor: \(6.69197\)
Root analytic conductor: \(6.69197\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (1081, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (0:\ ),\ 0.986 - 0.164i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8514767959 - 0.07070572273i\)
\(L(\frac12)\) \(\approx\) \(0.8514767959 - 0.07070572273i\)
\(L(1)\) \(\approx\) \(0.7601131412 - 0.1239439312i\)
\(L(1)\) \(\approx\) \(0.7601131412 - 0.1239439312i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (0.215 - 0.976i)T \)
3 \( 1 + (-0.262 + 0.964i)T \)
5 \( 1 + (-0.998 + 0.0483i)T \)
7 \( 1 + (0.715 + 0.698i)T \)
13 \( 1 + (-0.989 - 0.144i)T \)
17 \( 1 + (-0.970 - 0.239i)T \)
19 \( 1 + (0.644 - 0.764i)T \)
23 \( 1 + (-0.607 - 0.794i)T \)
29 \( 1 + (0.399 - 0.916i)T \)
31 \( 1 + (-0.607 + 0.794i)T \)
37 \( 1 + (0.981 - 0.192i)T \)
41 \( 1 + (-0.970 + 0.239i)T \)
43 \( 1 + (0.836 + 0.548i)T \)
47 \( 1 + (0.215 - 0.976i)T \)
53 \( 1 + (-0.809 + 0.587i)T \)
59 \( 1 + (-0.527 + 0.849i)T \)
61 \( 1 + T \)
67 \( 1 + (0.836 - 0.548i)T \)
71 \( 1 + (-0.607 + 0.794i)T \)
73 \( 1 + (0.309 - 0.951i)T \)
79 \( 1 + (-0.998 + 0.0483i)T \)
83 \( 1 + (0.981 + 0.192i)T \)
89 \( 1 + (-0.809 + 0.587i)T \)
97 \( 1 + (0.715 + 0.698i)T \)
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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

−20.51704942475936119613082153933, −19.913738426178554321150762923663, −19.086046199488936698226817362988, −18.33624400426977397649877084636, −17.59726960249843201065302943452, −17.03118183288836050022333254609, −16.29354659906445321106321522893, −15.47538251448851941554943308964, −14.48457069741682773045875986171, −14.15431838034349470332454685628, −13.148018591848504057056183565647, −12.4788905945308775623737342172, −11.70746788130124070017149136867, −11.0343139011917498168056415426, −9.79220409329756728073503799984, −8.62951940237339694369378169721, −7.95085943758178858740194244826, −7.3974450390746684688198093651, −6.904382006084422244137245819865, −5.82232011590009364195623001143, −4.92872517148802178724333569186, −4.21790576017831758187810886706, −3.23285405562670336982942166607, −1.80706134553775944388046965533, −0.556682106157914381106255951635, 0.63093756928433105548189929928, 2.30540762219904073663291925252, 2.922572053023809101822372387577, 4.00605732557645305585559194350, 4.71976382328208948208082943867, 5.11469041301798867221575424149, 6.289533170709921568461642763144, 7.66376528915395043840353419146, 8.59734784605416233983366417025, 9.15214451235911733428741115045, 10.06945762446510309553240404066, 10.89363542455036390042108916656, 11.5367776244980711241088633777, 11.95398859809849398684216658462, 12.761164788262932991681412802662, 14.02078151208151419458183460803, 14.70420984849810264019567617424, 15.31343315756327673690799095070, 15.94551995058387617057300525820, 17.05931553803065677903081322302, 17.87113279814129775343757705680, 18.44810752200783260798827212156, 19.567236987488699906470852510190, 20.03122006766563647231486118222, 20.62872379850015131227005707529

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