↑

Properties

Label 1-33e2-1089.13-r1-0-0
Degree $1$
Conductor $1089$
Sign $0.999 + 0.0175i$
Analytic cond. $117.029$
Root an. cond. $117.029$
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  + (−0.00951 + 0.999i)2-s + (−0.999 − 0.0190i)4-s + (−0.761 − 0.647i)5-s + (0.0665 − 0.997i)7-s + (0.0285 − 0.999i)8-s + (0.654 − 0.755i)10-s + (−0.820 + 0.572i)13-s + (0.997 + 0.0760i)14-s + (0.999 + 0.0380i)16-s + (0.998 − 0.0570i)17-s + (0.254 + 0.967i)19-s + (0.749 + 0.662i)20-s + (0.928 − 0.371i)23-s + (0.161 + 0.986i)25-s + (−0.564 − 0.825i)26-s + ⋯
L(s)  = 1  + (−0.00951 + 0.999i)2-s + (−0.999 − 0.0190i)4-s + (−0.761 − 0.647i)5-s + (0.0665 − 0.997i)7-s + (0.0285 − 0.999i)8-s + (0.654 − 0.755i)10-s + (−0.820 + 0.572i)13-s + (0.997 + 0.0760i)14-s + (0.999 + 0.0380i)16-s + (0.998 − 0.0570i)17-s + (0.254 + 0.967i)19-s + (0.749 + 0.662i)20-s + (0.928 − 0.371i)23-s + (0.161 + 0.986i)25-s + (−0.564 − 0.825i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0175i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0175i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $0.999 + 0.0175i$
Analytic conductor: \(117.029\)
Root analytic conductor: \(117.029\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (1:\ ),\ 0.999 + 0.0175i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.280058308 + 0.01126322585i\)
\(L(\frac12)\) \(\approx\) \(1.280058308 + 0.01126322585i\)
\(L(1)\) \(\approx\) \(0.8111148092 + 0.2026139916i\)
\(L(1)\) \(\approx\) \(0.8111148092 + 0.2026139916i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.00951 + 0.999i)T \)
5 \( 1 + (-0.761 - 0.647i)T \)
7 \( 1 + (0.0665 - 0.997i)T \)
13 \( 1 + (-0.820 + 0.572i)T \)
17 \( 1 + (0.998 - 0.0570i)T \)
19 \( 1 + (0.254 + 0.967i)T \)
23 \( 1 + (0.928 - 0.371i)T \)
29 \( 1 + (0.935 - 0.353i)T \)
31 \( 1 + (-0.683 - 0.730i)T \)
37 \( 1 + (-0.921 - 0.389i)T \)
41 \( 1 + (0.217 - 0.976i)T \)
43 \( 1 + (-0.235 + 0.971i)T \)
47 \( 1 + (-0.398 + 0.917i)T \)
53 \( 1 + (-0.466 + 0.884i)T \)
59 \( 1 + (0.953 + 0.299i)T \)
61 \( 1 + (-0.861 + 0.508i)T \)
67 \( 1 + (-0.995 + 0.0950i)T \)
71 \( 1 + (-0.362 + 0.931i)T \)
73 \( 1 + (0.985 - 0.170i)T \)
79 \( 1 + (0.991 - 0.132i)T \)
83 \( 1 + (0.969 - 0.244i)T \)
89 \( 1 + (0.841 + 0.540i)T \)
97 \( 1 + (-0.761 + 0.647i)T \)
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

−21.37764332607961976152879447703, −20.39286182696145891515664310707, −19.41820949982580646138641652699, −19.26623292671184940124408026324, −18.21780876847303729386205902782, −17.79451655126751333013591315302, −16.68066560731237862660875187423, −15.532964069646207957348879832861, −14.90876483528259804722063117947, −14.242213258575933904557925855420, −13.14107397291433525291854009671, −12.23584852035152087920197063850, −11.87635583056138406299016279730, −10.974753625520395049877826403703, −10.245606142412793668611773970802, −9.3490121054257989887878643644, −8.50137443688661047614280673931, −7.70832441248156663106137509871, −6.69544668244239139844197268297, −5.28544435987514335954806800939, −4.86032140349908942329816500746, −3.34211195673838457362275287787, −3.058939749232557742782385101314, −1.99428701923381333146117742646, −0.65755552661605318170965342304, 0.454406183483092521532524375651, 1.34781950907046875961618477621, 3.271712653180207690545387620576, 4.15419861654407785084568006929, 4.77048301817993333882767545752, 5.67977030724730214490516293597, 6.85319885646246207643504125609, 7.55258814802237467113306013538, 8.054241485644107933532678005245, 9.100752173101066523306408049943, 9.843449842982580701848943614705, 10.78495703991443522517624575819, 12.03323314621367366476195687136, 12.588819423619062420401504553837, 13.56888751712116062618195742122, 14.35194315915045300996323777252, 14.929778917226899305699147583253, 16.00414721561378438427776069274, 16.60517585927352914723110607972, 16.99468197564244318497606250563, 17.91108516265176208884353148303, 19.14256241357543952475140502775, 19.31557661263318563899460633075, 20.58023782388493688660350201821, 21.11761826953611425529586958871

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