Properties

Label 1-33e2-1089.155-r1-0-0
Degree $1$
Conductor $1089$
Sign $-0.992 + 0.122i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.723 − 0.690i)2-s + (0.0475 + 0.998i)4-s + (−0.981 + 0.189i)5-s + (0.580 − 0.814i)7-s + (0.654 − 0.755i)8-s + (0.841 + 0.540i)10-s + (0.0475 + 0.998i)13-s + (−0.981 + 0.189i)14-s + (−0.995 + 0.0950i)16-s + (0.142 − 0.989i)17-s + (−0.142 − 0.989i)19-s + (−0.235 − 0.971i)20-s + (−0.580 − 0.814i)23-s + (0.928 − 0.371i)25-s + (0.654 − 0.755i)26-s + ⋯
L(s)  = 1  + (−0.723 − 0.690i)2-s + (0.0475 + 0.998i)4-s + (−0.981 + 0.189i)5-s + (0.580 − 0.814i)7-s + (0.654 − 0.755i)8-s + (0.841 + 0.540i)10-s + (0.0475 + 0.998i)13-s + (−0.981 + 0.189i)14-s + (−0.995 + 0.0950i)16-s + (0.142 − 0.989i)17-s + (−0.142 − 0.989i)19-s + (−0.235 − 0.971i)20-s + (−0.580 − 0.814i)23-s + (0.928 − 0.371i)25-s + (0.654 − 0.755i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.992 + 0.122i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.992 + 0.122i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $-0.992 + 0.122i$
Analytic conductor: \(117.029\)
Root analytic conductor: \(117.029\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (1:\ ),\ -0.992 + 0.122i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.03544774184 - 0.5775148243i\)
\(L(\frac12)\) \(\approx\) \(-0.03544774184 - 0.5775148243i\)
\(L(1)\) \(\approx\) \(0.5762513208 - 0.2630639567i\)
\(L(1)\) \(\approx\) \(0.5762513208 - 0.2630639567i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.723 - 0.690i)T \)
5 \( 1 + (-0.981 + 0.189i)T \)
7 \( 1 + (0.580 - 0.814i)T \)
13 \( 1 + (0.0475 + 0.998i)T \)
17 \( 1 + (0.142 - 0.989i)T \)
19 \( 1 + (-0.142 - 0.989i)T \)
23 \( 1 + (-0.580 - 0.814i)T \)
29 \( 1 + (0.786 - 0.618i)T \)
31 \( 1 + (-0.888 + 0.458i)T \)
37 \( 1 + (0.841 + 0.540i)T \)
41 \( 1 + (-0.235 + 0.971i)T \)
43 \( 1 + (0.981 + 0.189i)T \)
47 \( 1 + (-0.235 - 0.971i)T \)
53 \( 1 + (-0.415 - 0.909i)T \)
59 \( 1 + (-0.723 + 0.690i)T \)
61 \( 1 + (0.235 + 0.971i)T \)
67 \( 1 + (0.235 - 0.971i)T \)
71 \( 1 + (0.142 + 0.989i)T \)
73 \( 1 + (0.415 - 0.909i)T \)
79 \( 1 + (-0.327 + 0.945i)T \)
83 \( 1 + (-0.580 + 0.814i)T \)
89 \( 1 + (0.142 - 0.989i)T \)
97 \( 1 + (0.981 + 0.189i)T \)
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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

−21.690635976222990366903558903304, −20.57187490906551174731566327922, −19.968885030422430526434868916393, −19.124303890940893230179120570899, −18.558490202220283829648532296271, −17.721589082704092437464135293526, −17.06090998094950780509956323752, −15.98500750315679174362189599144, −15.60534285348457860715249742015, −14.78420364172150350193596657111, −14.2620720705442261772148420227, −12.80665259314820047046534206792, −12.160551816801436053047171894619, −11.12437050635910237435109919215, −10.55198937426884736918255688852, −9.441866770816388106673772905703, −8.5726930036246415287784095915, −7.950967619207385523391480039554, −7.48399534078826774167889692076, −6.08937724941683321950437629580, −5.55053050576142294643141138052, −4.53525257367011941581450047150, −3.454030954717265215784341365021, −2.06020180720373338294572831446, −1.04770787079769020212103716839, 0.200286290538432137549392892516, 1.02722094034523134982936929866, 2.26914302083304793975861303002, 3.233839364188318209248870851648, 4.29129106974505609621441166409, 4.67574316710493520037724930199, 6.64708367128679936378355496458, 7.22559205708424769822418584148, 8.025351156027021082615903027, 8.750568364910979292672490545948, 9.71040417574802509601449977172, 10.60147130391621713156064514234, 11.41535412624270088356331185964, 11.697831782379470878543851691252, 12.74523056271619497658552189337, 13.725788571667713942693327524332, 14.46296787588475494274166116213, 15.6008450378442974660069300171, 16.41084948986081994228653260191, 16.89482857175359305330877929023, 18.06755299107066193759629744912, 18.41139450730429941799327790114, 19.53358916978602591853134646838, 19.85893925555201601124924738683, 20.67716222372994432234918046649

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