Properties

Label 1-1441-1441.1099-r0-0-0
Degree $1$
Conductor $1441$
Sign $0.969 - 0.243i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.748 + 0.663i)2-s + (−0.354 − 0.935i)3-s + (0.120 − 0.992i)4-s + (−0.354 − 0.935i)5-s + (0.885 + 0.464i)6-s + (−0.885 + 0.464i)7-s + (0.568 + 0.822i)8-s + (−0.748 + 0.663i)9-s + (0.885 + 0.464i)10-s + (−0.970 + 0.239i)12-s + (−0.885 + 0.464i)13-s + (0.354 − 0.935i)14-s + (−0.748 + 0.663i)15-s + (−0.970 − 0.239i)16-s + (−0.970 − 0.239i)17-s + (0.120 − 0.992i)18-s + ⋯
L(s)  = 1  + (−0.748 + 0.663i)2-s + (−0.354 − 0.935i)3-s + (0.120 − 0.992i)4-s + (−0.354 − 0.935i)5-s + (0.885 + 0.464i)6-s + (−0.885 + 0.464i)7-s + (0.568 + 0.822i)8-s + (−0.748 + 0.663i)9-s + (0.885 + 0.464i)10-s + (−0.970 + 0.239i)12-s + (−0.885 + 0.464i)13-s + (0.354 − 0.935i)14-s + (−0.748 + 0.663i)15-s + (−0.970 − 0.239i)16-s + (−0.970 − 0.239i)17-s + (0.120 − 0.992i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.969 - 0.243i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.969 - 0.243i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $0.969 - 0.243i$
Analytic conductor: \(6.69197\)
Root analytic conductor: \(6.69197\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (1099, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (0:\ ),\ 0.969 - 0.243i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2929984209 - 0.03626265879i\)
\(L(\frac12)\) \(\approx\) \(0.2929984209 - 0.03626265879i\)
\(L(1)\) \(\approx\) \(0.4149098362 - 0.04624452500i\)
\(L(1)\) \(\approx\) \(0.4149098362 - 0.04624452500i\)

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 \( 1 + (-0.354 - 0.935i)T \)
5 \( 1 + (-0.354 - 0.935i)T \)
7 \( 1 + (-0.885 + 0.464i)T \)
13 \( 1 + (-0.885 + 0.464i)T \)
17 \( 1 + (-0.970 - 0.239i)T \)
19 \( 1 + (-0.970 + 0.239i)T \)
23 \( 1 + (-0.568 + 0.822i)T \)
29 \( 1 + (-0.748 - 0.663i)T \)
31 \( 1 + (-0.568 - 0.822i)T \)
37 \( 1 + (-0.120 + 0.992i)T \)
41 \( 1 + (0.970 - 0.239i)T \)
43 \( 1 + (0.354 + 0.935i)T \)
47 \( 1 + (0.748 - 0.663i)T \)
53 \( 1 + T \)
59 \( 1 + (-0.970 - 0.239i)T \)
61 \( 1 - T \)
67 \( 1 + (0.354 - 0.935i)T \)
71 \( 1 + (-0.568 - 0.822i)T \)
73 \( 1 + T \)
79 \( 1 + (-0.354 - 0.935i)T \)
83 \( 1 + (0.120 + 0.992i)T \)
89 \( 1 + T \)
97 \( 1 + (-0.885 + 0.464i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−20.5787466075064585901615742688, −19.82480249797504046086497216185, −19.52440003774259232435182758866, −18.507705775087046593461095756092, −17.714030130895455050047516906989, −17.09464728581993644015470963633, −16.30481862927916895832246083726, −15.659709752310057523124988372598, −14.88833337185439155420246647576, −13.98765390729842658361015432585, −12.73538918382343498855932371390, −12.27685875867647359429823408096, −11.08339358829948761536693323600, −10.691299329757395876147910159755, −10.19187583579355829340393174475, −9.31304719608240788075493071504, −8.623314608276151932078223952331, −7.40995702269512839291509822140, −6.814248608345498455085069005125, −5.87998425476165617493363518359, −4.37781359393306255220075016966, −3.89139766347194259779889146658, −2.95615245667750892595076178570, −2.287403227039650266465144718396, −0.36116474780730765838631219840, 0.41806236738628506571489447674, 1.766029014688566864676578287295, 2.420765310211746832286505259533, 4.14630362589189973178229269826, 5.13728002859868480683983579609, 5.95312719393391155966165207026, 6.56874370216905869650340900732, 7.51105856996092330127272150034, 8.057997310087569298673228435176, 9.13087365574721770048486538719, 9.41100214903183877699570891697, 10.667768672100760520378719430477, 11.629517708305382610366734025051, 12.22224703208301492918274018541, 13.1418751084430992275774850365, 13.672846882158331750251312272567, 14.89872823082608448802978331564, 15.560283060998279516408308097961, 16.453300259815724789409426027864, 16.9165089434520029056451858018, 17.546549831217836867006507344589, 18.5044944756364929440221513516, 19.10105040950941422520600823213, 19.72290041761485685233835027974, 20.16843992790602457980677637854

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