Properties

Label 1-1441-1441.1124-r0-0-0
Degree $1$
Conductor $1441$
Sign $-0.957 + 0.287i$
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  + (−0.748 + 0.663i)2-s + (−0.262 + 0.964i)3-s + (0.120 − 0.992i)4-s + (−0.998 + 0.0483i)5-s + (−0.443 − 0.896i)6-s + (0.989 + 0.144i)7-s + (0.568 + 0.822i)8-s + (−0.861 − 0.506i)9-s + (0.715 − 0.698i)10-s + (0.926 + 0.377i)12-s + (0.168 + 0.985i)13-s + (−0.836 + 0.548i)14-s + (0.215 − 0.976i)15-s + (−0.970 − 0.239i)16-s + (0.644 + 0.764i)17-s + (0.981 − 0.192i)18-s + ⋯
L(s)  = 1  + (−0.748 + 0.663i)2-s + (−0.262 + 0.964i)3-s + (0.120 − 0.992i)4-s + (−0.998 + 0.0483i)5-s + (−0.443 − 0.896i)6-s + (0.989 + 0.144i)7-s + (0.568 + 0.822i)8-s + (−0.861 − 0.506i)9-s + (0.715 − 0.698i)10-s + (0.926 + 0.377i)12-s + (0.168 + 0.985i)13-s + (−0.836 + 0.548i)14-s + (0.215 − 0.976i)15-s + (−0.970 − 0.239i)16-s + (0.644 + 0.764i)17-s + (0.981 − 0.192i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.957 + 0.287i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.957 + 0.287i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $-0.957 + 0.287i$
Analytic conductor: \(6.69197\)
Root analytic conductor: \(6.69197\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (1124, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (0:\ ),\ -0.957 + 0.287i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1144982697 + 0.7795125478i\)
\(L(\frac12)\) \(\approx\) \(0.1144982697 + 0.7795125478i\)
\(L(1)\) \(\approx\) \(0.4894658096 + 0.4363525074i\)
\(L(1)\) \(\approx\) \(0.4894658096 + 0.4363525074i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (-0.748 + 0.663i)T \)
3 \( 1 + (-0.262 + 0.964i)T \)
5 \( 1 + (-0.998 + 0.0483i)T \)
7 \( 1 + (0.989 + 0.144i)T \)
13 \( 1 + (0.168 + 0.985i)T \)
17 \( 1 + (0.644 + 0.764i)T \)
19 \( 1 + (-0.0724 + 0.997i)T \)
23 \( 1 + (-0.958 - 0.285i)T \)
29 \( 1 + (0.215 + 0.976i)T \)
31 \( 1 + (0.607 - 0.794i)T \)
37 \( 1 + (0.681 - 0.732i)T \)
41 \( 1 + (0.527 + 0.849i)T \)
43 \( 1 + (0.262 - 0.964i)T \)
47 \( 1 + (-0.399 - 0.916i)T \)
53 \( 1 + T \)
59 \( 1 + (-0.970 - 0.239i)T \)
61 \( 1 + (0.809 + 0.587i)T \)
67 \( 1 + (0.262 + 0.964i)T \)
71 \( 1 + (0.943 + 0.331i)T \)
73 \( 1 + T \)
79 \( 1 + (-0.262 + 0.964i)T \)
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

−20.16706113398427454436284820505, −19.59351613492422245922528520630, −18.91930791048793504219560476658, −18.02826158033167095414900598781, −17.7623188580980556331710582375, −16.89938755082145033459393690759, −16.02183909003138941550110312424, −15.23627382968875425809680775113, −14.07151138267077403890956344853, −13.35521030843371340128195728497, −12.35706804806366621502481731347, −11.93508241370308256308926757181, −11.19168067128821533372816495418, −10.71172755605686786797252312327, −9.52281049511417987367522499827, −8.3750093112995515756638500976, −7.95067508478299232075330260946, −7.47704833283513823809027037309, −6.5097848168479683481266212264, −5.20348856657103117345315100100, −4.332611567891147276480066567551, −3.153899698165779619560397752898, −2.416977453564626192559214590298, −1.19534978278539753415812894353, −0.54433281645047651772119626317, 1.04982776696061226390282621573, 2.26016150283629019038065152057, 3.84393122667215088591517143564, 4.33239058225638805765321164025, 5.32133243339885408744855315556, 6.06429055035893784915753848520, 7.08620730706322395483132835924, 8.21409109319518813141429754404, 8.33640971449605424938032645713, 9.41084575756726148135249356824, 10.26864340412709549442398245152, 10.95415451213060361586132592365, 11.61594142253110095316057645085, 12.28220115244292623498386052721, 14.04193057135670820009522098582, 14.58267106671254971372112513141, 15.080461747882438778072305247462, 15.92670314332653069341835859936, 16.55513641753458030744055472634, 17.003004105689053767954552245410, 18.09116291017140235761187444799, 18.61909251296952835855373233857, 19.54472165672737896985841213688, 20.23451817287001161774415220100, 20.99212356894260196659642087721

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