Properties

Label 1-33e2-1089.157-r0-0-0
Degree $1$
Conductor $1089$
Sign $0.573 - 0.819i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.953 − 0.299i)2-s + (0.820 − 0.572i)4-s + (−0.851 + 0.524i)5-s + (−0.532 − 0.846i)7-s + (0.610 − 0.791i)8-s + (−0.654 + 0.755i)10-s + (0.797 + 0.603i)13-s + (−0.761 − 0.647i)14-s + (0.345 − 0.938i)16-s + (−0.254 + 0.967i)17-s + (−0.362 + 0.931i)19-s + (−0.398 + 0.917i)20-s + (0.928 − 0.371i)23-s + (0.449 − 0.893i)25-s + (0.941 + 0.336i)26-s + ⋯
L(s)  = 1  + (0.953 − 0.299i)2-s + (0.820 − 0.572i)4-s + (−0.851 + 0.524i)5-s + (−0.532 − 0.846i)7-s + (0.610 − 0.791i)8-s + (−0.654 + 0.755i)10-s + (0.797 + 0.603i)13-s + (−0.761 − 0.647i)14-s + (0.345 − 0.938i)16-s + (−0.254 + 0.967i)17-s + (−0.362 + 0.931i)19-s + (−0.398 + 0.917i)20-s + (0.928 − 0.371i)23-s + (0.449 − 0.893i)25-s + (0.941 + 0.336i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.573 - 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.573 - 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $0.573 - 0.819i$
Analytic conductor: \(5.05729\)
Root analytic conductor: \(5.05729\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (0:\ ),\ 0.573 - 0.819i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.087037662 - 1.086634356i\)
\(L(\frac12)\) \(\approx\) \(2.087037662 - 1.086634356i\)
\(L(1)\) \(\approx\) \(1.593687175 - 0.4184607132i\)
\(L(1)\) \(\approx\) \(1.593687175 - 0.4184607132i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.953 - 0.299i)T \)
5 \( 1 + (-0.851 + 0.524i)T \)
7 \( 1 + (-0.532 - 0.846i)T \)
13 \( 1 + (0.797 + 0.603i)T \)
17 \( 1 + (-0.254 + 0.967i)T \)
19 \( 1 + (-0.362 + 0.931i)T \)
23 \( 1 + (0.928 - 0.371i)T \)
29 \( 1 + (0.548 - 0.836i)T \)
31 \( 1 + (0.483 - 0.875i)T \)
37 \( 1 + (0.974 - 0.226i)T \)
41 \( 1 + (0.749 - 0.662i)T \)
43 \( 1 + (0.235 - 0.971i)T \)
47 \( 1 + (0.861 - 0.508i)T \)
53 \( 1 + (-0.985 - 0.170i)T \)
59 \( 1 + (-0.948 + 0.318i)T \)
61 \( 1 + (-0.217 - 0.976i)T \)
67 \( 1 + (-0.995 + 0.0950i)T \)
71 \( 1 + (0.774 + 0.633i)T \)
73 \( 1 + (0.696 - 0.717i)T \)
79 \( 1 + (-0.432 - 0.901i)T \)
83 \( 1 + (-0.0665 + 0.997i)T \)
89 \( 1 + (0.841 + 0.540i)T \)
97 \( 1 + (-0.851 - 0.524i)T \)
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

−21.57396839948476200284151221781, −20.95702548102486609961900779520, −19.99373662133972497020674758580, −19.56925217379854865316376586320, −18.47965639705507945590150707507, −17.56204119706051677423827714752, −16.49621117096546248282585154806, −15.81155973823555048886103061285, −15.50595852935564545729765223771, −14.644379735993503733601548036001, −13.52873256850388877024585637726, −12.86789592527572246359057123956, −12.3057425555509872676835007534, −11.39622023641326761989815843631, −10.866414353353143851057918434867, −9.30076413427878003063830754528, −8.62203941396030418917648052399, −7.732663036323709657838444330274, −6.83299571211976253413958435864, −6.00445563059092048836707126973, −5.01984498108469250350963376471, −4.45315148302379245013596952466, −3.15651877634986022436398505943, −2.81019365479197869246511104519, −1.13606341225004118022444245154, 0.83481690158960671448376214914, 2.15056017405755469969426118245, 3.25326227066399329876522769147, 4.01700792885017947045778692620, 4.407132501786952762981400700751, 5.99833784685134145417708839456, 6.493875268793956081717652045640, 7.373222507605246139868463534532, 8.229164532993133295245386993579, 9.558348176908415308528331028639, 10.714629231946741651991744113640, 10.82908216068082371354777144649, 11.926314913140799233348771918930, 12.64839136354036651146868467741, 13.48664537859646284196217395691, 14.177055411201661423968296979567, 15.01107901156331534867887882994, 15.668512887431691315883372681191, 16.45748579871991445169329793075, 17.18298302244380069428839425974, 18.74205945737193635958349794883, 19.05473958589425717451931681518, 19.84246078766958071853253988043, 20.617589527749305957278087783171, 21.28773001983886308297700712928

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