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Properties

Label 1-33e2-1089.1048-r0-0-0
Degree $1$
Conductor $1089$
Sign $-0.329 + 0.944i$
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  + (0.964 + 0.263i)2-s + (0.861 + 0.508i)4-s + (0.640 + 0.768i)5-s + (−0.290 + 0.956i)7-s + (0.696 + 0.717i)8-s + (0.415 + 0.909i)10-s + (−0.217 + 0.976i)13-s + (−0.532 + 0.846i)14-s + (0.483 + 0.875i)16-s + (−0.0285 − 0.999i)17-s + (0.610 − 0.791i)19-s + (0.161 + 0.986i)20-s + (−0.327 + 0.945i)23-s + (−0.179 + 0.983i)25-s + (−0.466 + 0.884i)26-s + ⋯
L(s)  = 1  + (0.964 + 0.263i)2-s + (0.861 + 0.508i)4-s + (0.640 + 0.768i)5-s + (−0.290 + 0.956i)7-s + (0.696 + 0.717i)8-s + (0.415 + 0.909i)10-s + (−0.217 + 0.976i)13-s + (−0.532 + 0.846i)14-s + (0.483 + 0.875i)16-s + (−0.0285 − 0.999i)17-s + (0.610 − 0.791i)19-s + (0.161 + 0.986i)20-s + (−0.327 + 0.945i)23-s + (−0.179 + 0.983i)25-s + (−0.466 + 0.884i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.329 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.329 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $-0.329 + 0.944i$
Analytic conductor: \(5.05729\)
Root analytic conductor: \(5.05729\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (1048, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (0:\ ),\ -0.329 + 0.944i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.764715710 + 2.484258517i\)
\(L(\frac12)\) \(\approx\) \(1.764715710 + 2.484258517i\)
\(L(1)\) \(\approx\) \(1.758757799 + 1.004774443i\)
\(L(1)\) \(\approx\) \(1.758757799 + 1.004774443i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.964 + 0.263i)T \)
5 \( 1 + (0.640 + 0.768i)T \)
7 \( 1 + (-0.290 + 0.956i)T \)
13 \( 1 + (-0.217 + 0.976i)T \)
17 \( 1 + (-0.0285 - 0.999i)T \)
19 \( 1 + (0.610 - 0.791i)T \)
23 \( 1 + (-0.327 + 0.945i)T \)
29 \( 1 + (-0.761 + 0.647i)T \)
31 \( 1 + (-0.595 - 0.803i)T \)
37 \( 1 + (0.198 - 0.980i)T \)
41 \( 1 + (0.988 - 0.151i)T \)
43 \( 1 + (0.928 + 0.371i)T \)
47 \( 1 + (0.449 - 0.893i)T \)
53 \( 1 + (0.516 + 0.856i)T \)
59 \( 1 + (-0.625 + 0.780i)T \)
61 \( 1 + (-0.710 - 0.703i)T \)
67 \( 1 + (-0.888 - 0.458i)T \)
71 \( 1 + (-0.564 - 0.825i)T \)
73 \( 1 + (0.0855 + 0.996i)T \)
79 \( 1 + (-0.830 + 0.556i)T \)
83 \( 1 + (0.797 - 0.603i)T \)
89 \( 1 + (-0.959 - 0.281i)T \)
97 \( 1 + (0.640 - 0.768i)T \)
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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

−21.01179714851981143986788677593, −20.58697265616328549354342691700, −19.95784839603941962515140229238, −19.215751384244656775474229116177, −18.01653543628612604128096758408, −17.08879789860275238430210436480, −16.49535552243660905811051948464, −15.738603031195451700171574481191, −14.66545357673575818099371280366, −14.06667499037004928489496397063, −13.16161773295560339748249287371, −12.76163238793981845320431112644, −12.00327393386949344487519451643, −10.71749229849560214106196125740, −10.29022388732406507867463790109, −9.46491909209529409620692106245, −8.15453312121177636335571269129, −7.35747844063387334890847471489, −6.17736016951420806249962937339, −5.69279321627897945324559161112, −4.64287834109814279261840934329, −3.92839507497154347750807077355, −2.94631956452160033985452963283, −1.78817794704817934133906647460, −0.88320638465875832285373798261, 1.84626887013979361552321557149, 2.555733966150731919057508368892, 3.32072518100183559202798004872, 4.45714653969335393419752252988, 5.56743991616795046630489991018, 5.93219482863071661277997810461, 7.09586610141992000700313920275, 7.44493812919275816250308447441, 9.10179351056810247420136756606, 9.49660977673896655877406060156, 10.88104767447448259147555684904, 11.4907686152199040079032122220, 12.22795130051268712108025701316, 13.216320142594657671820758467293, 13.87377682426797282747129022294, 14.53924214559326725501206325801, 15.320714818734740744222775708384, 16.01166441168476532256657749485, 16.81587397761889706239963955683, 17.851802202787937386529651646300, 18.480213390630009003087870512054, 19.41366321296036815336331016667, 20.269309495366312611160398989824, 21.4059805811241419810220136433, 21.62347159089742991720509266294

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