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Properties

Label 1-1441-1441.1059-r0-0-0
Degree $1$
Conductor $1441$
Sign $0.941 + 0.337i$
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  + (0.120 + 0.992i)2-s + (−0.861 + 0.506i)3-s + (−0.970 + 0.239i)4-s + (0.995 + 0.0965i)5-s + (−0.607 − 0.794i)6-s + (0.958 − 0.285i)7-s + (−0.354 − 0.935i)8-s + (0.485 − 0.873i)9-s + (0.0241 + 0.999i)10-s + (0.715 − 0.698i)12-s + (−0.943 − 0.331i)13-s + (0.399 + 0.916i)14-s + (−0.906 + 0.421i)15-s + (0.885 − 0.464i)16-s + (−0.168 − 0.985i)17-s + (0.926 + 0.377i)18-s + ⋯
L(s)  = 1  + (0.120 + 0.992i)2-s + (−0.861 + 0.506i)3-s + (−0.970 + 0.239i)4-s + (0.995 + 0.0965i)5-s + (−0.607 − 0.794i)6-s + (0.958 − 0.285i)7-s + (−0.354 − 0.935i)8-s + (0.485 − 0.873i)9-s + (0.0241 + 0.999i)10-s + (0.715 − 0.698i)12-s + (−0.943 − 0.331i)13-s + (0.399 + 0.916i)14-s + (−0.906 + 0.421i)15-s + (0.885 − 0.464i)16-s + (−0.168 − 0.985i)17-s + (0.926 + 0.377i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.941 + 0.337i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.941 + 0.337i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $0.941 + 0.337i$
Analytic conductor: \(6.69197\)
Root analytic conductor: \(6.69197\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (1059, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (0:\ ),\ 0.941 + 0.337i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.141159480 + 0.1984603667i\)
\(L(\frac12)\) \(\approx\) \(1.141159480 + 0.1984603667i\)
\(L(1)\) \(\approx\) \(0.8426775256 + 0.4022268105i\)
\(L(1)\) \(\approx\) \(0.8426775256 + 0.4022268105i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (0.120 + 0.992i)T \)
3 \( 1 + (-0.861 + 0.506i)T \)
5 \( 1 + (0.995 + 0.0965i)T \)
7 \( 1 + (0.958 - 0.285i)T \)
13 \( 1 + (-0.943 - 0.331i)T \)
17 \( 1 + (-0.168 - 0.985i)T \)
19 \( 1 + (-0.989 + 0.144i)T \)
23 \( 1 + (0.836 - 0.548i)T \)
29 \( 1 + (-0.906 - 0.421i)T \)
31 \( 1 + (-0.262 + 0.964i)T \)
37 \( 1 + (-0.0724 + 0.997i)T \)
41 \( 1 + (-0.443 - 0.896i)T \)
43 \( 1 + (-0.861 + 0.506i)T \)
47 \( 1 + (-0.681 - 0.732i)T \)
53 \( 1 + T \)
59 \( 1 + (0.885 - 0.464i)T \)
61 \( 1 + (0.309 - 0.951i)T \)
67 \( 1 + (-0.861 - 0.506i)T \)
71 \( 1 + (0.779 - 0.626i)T \)
73 \( 1 + T \)
79 \( 1 + (-0.861 + 0.506i)T \)
83 \( 1 + (0.926 - 0.377i)T \)
89 \( 1 + (0.309 - 0.951i)T \)
97 \( 1 + (0.0241 - 0.999i)T \)
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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

−20.993501052254064119682740503279, −19.86898298333854329950160161062, −19.10815316719463322716989236772, −18.42174798302754321045450172340, −17.71167185721686839265730558007, −17.19159432323950524224469729880, −16.70032908800700303539329748056, −14.92122166870042145592286054516, −14.62527293103605164131580652347, −13.40336083744242762156114362071, −13.029569526670764115611028806724, −12.24539217627573888175335618162, −11.43902727292926989040203012363, −10.83362111058570730079885644564, −10.137712556213187522275508472644, −9.22037332107714736253857700821, −8.415396236407463695814371764542, −7.361935502326677920110841516897, −6.2619744480054906548812239846, −5.423638865303004542666435536814, −4.93539357668474324445776122159, −4.0035860885991290256361381503, −2.36994653866846079547947353735, −1.96728800305634586223330166236, −1.12095802253006152511329619047, 0.52146813400622116104745620684, 1.8793116867132396430222121561, 3.28851742866783121710206981344, 4.56065270703282770858945006137, 5.01675535418793159482033981742, 5.58775160298052000667747366549, 6.69289764068399776460331450503, 7.086317313138654771571607008104, 8.30312172441487905005180381363, 9.14625907087626260585493212962, 9.95743358388675193206730414226, 10.56300855354565657825219417282, 11.54669951423583379305680284681, 12.511832266982551628628117105579, 13.258331973149318672356529011216, 14.154633327218139603085187335798, 14.86242949818809141017854146418, 15.31680177881361262101809221452, 16.625694632670510475413785545674, 16.87202947485253464898108435860, 17.56623013937165727922396612665, 18.122491931535830235781041240774, 18.81014255996988903434099869850, 20.29624796740833388783793386719, 21.14055167685655703881617130172

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