Properties

Label 1-33e2-1089.25-r0-0-0
Degree $1$
Conductor $1089$
Sign $0.996 + 0.0875i$
Analytic cond. $5.05729$
Root an. cond. $5.05729$
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  + (0.997 + 0.0760i)2-s + (0.988 + 0.151i)4-s + (0.797 − 0.603i)5-s + (0.861 + 0.508i)7-s + (0.974 + 0.226i)8-s + (0.841 − 0.540i)10-s + (0.161 + 0.986i)13-s + (0.820 + 0.572i)14-s + (0.953 + 0.299i)16-s + (0.897 − 0.441i)17-s + (−0.466 − 0.884i)19-s + (0.879 − 0.475i)20-s + (−0.995 − 0.0950i)23-s + (0.272 − 0.962i)25-s + (0.0855 + 0.996i)26-s + ⋯
L(s)  = 1  + (0.997 + 0.0760i)2-s + (0.988 + 0.151i)4-s + (0.797 − 0.603i)5-s + (0.861 + 0.508i)7-s + (0.974 + 0.226i)8-s + (0.841 − 0.540i)10-s + (0.161 + 0.986i)13-s + (0.820 + 0.572i)14-s + (0.953 + 0.299i)16-s + (0.897 − 0.441i)17-s + (−0.466 − 0.884i)19-s + (0.879 − 0.475i)20-s + (−0.995 − 0.0950i)23-s + (0.272 − 0.962i)25-s + (0.0855 + 0.996i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0875i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0875i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $0.996 + 0.0875i$
Analytic conductor: \(5.05729\)
Root analytic conductor: \(5.05729\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (0:\ ),\ 0.996 + 0.0875i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.904689630 + 0.1713288803i\)
\(L(\frac12)\) \(\approx\) \(3.904689630 + 0.1713288803i\)
\(L(1)\) \(\approx\) \(2.439832191 + 0.06880579777i\)
\(L(1)\) \(\approx\) \(2.439832191 + 0.06880579777i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.997 - 0.0760i)T \)
5 \( 1 + (-0.797 + 0.603i)T \)
7 \( 1 + (-0.861 - 0.508i)T \)
13 \( 1 + (-0.161 - 0.986i)T \)
17 \( 1 + (-0.897 + 0.441i)T \)
19 \( 1 + (0.466 + 0.884i)T \)
23 \( 1 + (0.995 + 0.0950i)T \)
29 \( 1 + (0.969 + 0.244i)T \)
31 \( 1 + (-0.964 - 0.263i)T \)
37 \( 1 + (0.998 + 0.0570i)T \)
41 \( 1 + (0.179 - 0.983i)T \)
43 \( 1 + (0.327 - 0.945i)T \)
47 \( 1 + (0.991 + 0.132i)T \)
53 \( 1 + (0.736 + 0.676i)T \)
59 \( 1 + (0.761 - 0.647i)T \)
61 \( 1 + (0.432 - 0.901i)T \)
67 \( 1 + (-0.723 + 0.690i)T \)
71 \( 1 + (0.985 - 0.170i)T \)
73 \( 1 + (-0.198 + 0.980i)T \)
79 \( 1 + (-0.483 + 0.875i)T \)
83 \( 1 + (0.398 + 0.917i)T \)
89 \( 1 + (0.142 + 0.989i)T \)
97 \( 1 + (-0.797 - 0.603i)T \)
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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

−21.4151072037068630971843965343, −20.76803252654103804914742816110, −20.29798132595534859415383343120, −19.12767605693631435271598465130, −18.381097773005427702337527653507, −17.31470412641199464906079158151, −16.91393296028506268370701377026, −15.67702901099076131239287726657, −14.93881160474224146984382032655, −14.21620323754500444348457794690, −13.80400833008912964253788782001, −12.82621063331859060732520031176, −12.08640586256932637638278229105, −11.05481775577990960393544836366, −10.43129274245352499151400538844, −9.89584951350685996642940873494, −8.238018992708550225717194470157, −7.61078048929901661131659845682, −6.59927571159819063552361623903, −5.73439145432559388200880062770, −5.20157278206447876401014372018, −3.95691730069307133405894087750, −3.2822115373934581800988666710, −2.09074853329473267224317984053, −1.409837407667605873837043501957, 1.47463030181426213768666417256, 2.06032374126652909565670218500, 3.1343892106699805835053334659, 4.49563953894588861738383574873, 4.88660173949484611809069763432, 5.85958751456014377263073388021, 6.50585820157469293642263789923, 7.666217691113713772923224527118, 8.51712432068303889178239097798, 9.46052535167194906814545107638, 10.42167284342793285209612423696, 11.52899355893609767191328005394, 11.933043124386505007578188413489, 12.89502880249752644250772699920, 13.67596307635715642095917752193, 14.27720897624593750098347990331, 14.97823726229186277649061098009, 15.9837997236710825085468348015, 16.63065769886428360501076603073, 17.42135709881720423021935280561, 18.25577527925052101498828164839, 19.27440684613863428084265239691, 20.21004780895667509506497095280, 21.11038223308100151116544639788, 21.26634796416131198774617650247

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