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Properties

Label 1-1441-1441.1075-r1-0-0
Degree $1$
Conductor $1441$
Sign $0.768 - 0.640i$
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  + (0.970 − 0.239i)2-s + (0.981 + 0.192i)3-s + (0.885 − 0.464i)4-s + (−0.681 − 0.732i)5-s + (0.998 − 0.0483i)6-s + (−0.779 − 0.626i)7-s + (0.748 − 0.663i)8-s + (0.926 + 0.377i)9-s + (−0.836 − 0.548i)10-s + (0.958 − 0.285i)12-s + (0.262 + 0.964i)13-s + (−0.906 − 0.421i)14-s + (−0.527 − 0.849i)15-s + (0.568 − 0.822i)16-s + (0.607 + 0.794i)17-s + (0.989 + 0.144i)18-s + ⋯
L(s)  = 1  + (0.970 − 0.239i)2-s + (0.981 + 0.192i)3-s + (0.885 − 0.464i)4-s + (−0.681 − 0.732i)5-s + (0.998 − 0.0483i)6-s + (−0.779 − 0.626i)7-s + (0.748 − 0.663i)8-s + (0.926 + 0.377i)9-s + (−0.836 − 0.548i)10-s + (0.958 − 0.285i)12-s + (0.262 + 0.964i)13-s + (−0.906 − 0.421i)14-s + (−0.527 − 0.849i)15-s + (0.568 − 0.822i)16-s + (0.607 + 0.794i)17-s + (0.989 + 0.144i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.768 - 0.640i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.768 - 0.640i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $0.768 - 0.640i$
Analytic conductor: \(154.856\)
Root analytic conductor: \(154.856\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (1075, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (1:\ ),\ 0.768 - 0.640i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(5.874498539 - 2.126480833i\)
\(L(\frac12)\) \(\approx\) \(5.874498539 - 2.126480833i\)
\(L(1)\) \(\approx\) \(2.506941180 - 0.5922570566i\)
\(L(1)\) \(\approx\) \(2.506941180 - 0.5922570566i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (0.970 - 0.239i)T \)
3 \( 1 + (0.981 + 0.192i)T \)
5 \( 1 + (-0.681 - 0.732i)T \)
7 \( 1 + (-0.779 - 0.626i)T \)
13 \( 1 + (0.262 + 0.964i)T \)
17 \( 1 + (0.607 + 0.794i)T \)
19 \( 1 + (0.943 + 0.331i)T \)
23 \( 1 + (0.215 + 0.976i)T \)
29 \( 1 + (0.527 - 0.849i)T \)
31 \( 1 + (0.995 - 0.0965i)T \)
37 \( 1 + (-0.168 - 0.985i)T \)
41 \( 1 + (-0.0241 + 0.999i)T \)
43 \( 1 + (-0.981 - 0.192i)T \)
47 \( 1 + (0.644 - 0.764i)T \)
53 \( 1 + T \)
59 \( 1 + (0.568 - 0.822i)T \)
61 \( 1 + (-0.309 + 0.951i)T \)
67 \( 1 + (0.981 - 0.192i)T \)
71 \( 1 + (-0.861 - 0.506i)T \)
73 \( 1 - T \)
79 \( 1 + (-0.981 - 0.192i)T \)
83 \( 1 + (0.989 - 0.144i)T \)
89 \( 1 + (0.309 - 0.951i)T \)
97 \( 1 + (0.836 - 0.548i)T \)
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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

−20.40265224757650058173015911153, −20.1664996308649758763694975027, −19.16482836152800896255911068267, −18.63018310083310511364524036877, −17.774966633877503808062927379959, −16.318264927928439164467637637697, −15.78983151319695038587701633345, −15.26228587818535273794198042228, −14.567872919740435831320379238208, −13.84135697871531301178671187781, −13.14722380573515583193440073569, −12.25057451659358556534849312950, −11.850873224879915435858930848351, −10.60439966539902412683855091790, −9.92061503153876698238809643074, −8.68281131758499163081975328287, −8.0206099221877101061613046092, −7.11876002037915317959182664063, −6.6643650585000837074348871606, −5.58551254416964841985666502068, −4.58263836189048306336810200934, −3.4649655266553677969053021082, −3.002094695559760820200130602907, −2.525159417628608002060103768545, −0.91982505450799828002442462256, 0.91503516926767967630176788933, 1.73091366013067165789159005969, 2.99432022707728946697379721189, 3.74941670566586913131910324254, 4.1390238374510324105995670904, 5.10954750638001216556079217548, 6.21407870181754931861457973079, 7.21851379417891719665367216106, 7.78850077962918178511899053479, 8.82921997322305165072363687545, 9.76866839200325001138913433311, 10.30670584998916204300790293647, 11.55055805195403472278286548698, 12.110040832039509760307993691514, 13.11579578707621310587079474472, 13.48705621472796744865435227175, 14.24495820070034699796175872171, 15.10334694038404105985369969467, 15.859473128513454670543719433780, 16.28958566065741651221400179251, 17.07542354440985963536024733168, 18.74509136510841802951028185066, 19.339091406816870499954918949345, 19.7968173135136209074117410259, 20.4550115389823314028325131776

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