Properties

Label 1-33e2-1089.158-r1-0-0
Degree $1$
Conductor $1089$
Sign $-0.965 + 0.258i$
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  + (0.217 + 0.976i)2-s + (−0.905 + 0.424i)4-s + (−0.879 + 0.475i)5-s + (0.999 + 0.0380i)7-s + (−0.610 − 0.791i)8-s + (−0.654 − 0.755i)10-s + (0.123 + 0.992i)13-s + (0.179 + 0.983i)14-s + (0.640 − 0.768i)16-s + (0.254 + 0.967i)17-s + (−0.362 − 0.931i)19-s + (0.595 − 0.803i)20-s + (0.786 − 0.618i)23-s + (0.548 − 0.836i)25-s + (−0.941 + 0.336i)26-s + ⋯
L(s)  = 1  + (0.217 + 0.976i)2-s + (−0.905 + 0.424i)4-s + (−0.879 + 0.475i)5-s + (0.999 + 0.0380i)7-s + (−0.610 − 0.791i)8-s + (−0.654 − 0.755i)10-s + (0.123 + 0.992i)13-s + (0.179 + 0.983i)14-s + (0.640 − 0.768i)16-s + (0.254 + 0.967i)17-s + (−0.362 − 0.931i)19-s + (0.595 − 0.803i)20-s + (0.786 − 0.618i)23-s + (0.548 − 0.836i)25-s + (−0.941 + 0.336i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.965 + 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.965 + 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $-0.965 + 0.258i$
Analytic conductor: \(117.029\)
Root analytic conductor: \(117.029\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (158, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (1:\ ),\ -0.965 + 0.258i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2364757609 + 1.797213327i\)
\(L(\frac12)\) \(\approx\) \(0.2364757609 + 1.797213327i\)
\(L(1)\) \(\approx\) \(0.7961583127 + 0.6979032026i\)
\(L(1)\) \(\approx\) \(0.7961583127 + 0.6979032026i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.217 + 0.976i)T \)
5 \( 1 + (-0.879 + 0.475i)T \)
7 \( 1 + (0.999 + 0.0380i)T \)
13 \( 1 + (0.123 + 0.992i)T \)
17 \( 1 + (0.254 + 0.967i)T \)
19 \( 1 + (-0.362 - 0.931i)T \)
23 \( 1 + (0.786 - 0.618i)T \)
29 \( 1 + (-0.449 + 0.893i)T \)
31 \( 1 + (-0.999 - 0.0190i)T \)
37 \( 1 + (0.974 + 0.226i)T \)
41 \( 1 + (0.948 - 0.318i)T \)
43 \( 1 + (0.723 - 0.690i)T \)
47 \( 1 + (-0.00951 + 0.999i)T \)
53 \( 1 + (0.985 - 0.170i)T \)
59 \( 1 + (-0.749 + 0.662i)T \)
61 \( 1 + (0.953 - 0.299i)T \)
67 \( 1 + (0.580 - 0.814i)T \)
71 \( 1 + (-0.774 + 0.633i)T \)
73 \( 1 + (0.696 + 0.717i)T \)
79 \( 1 + (0.997 - 0.0760i)T \)
83 \( 1 + (0.830 - 0.556i)T \)
89 \( 1 + (-0.841 + 0.540i)T \)
97 \( 1 + (0.879 + 0.475i)T \)
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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

−20.8147953146281931570107807824, −20.23560627799257980866914824735, −19.5281388952603893641950380673, −18.64450710871618882554944710465, −18.0227543244913360512646562968, −17.15408092643244729692095275719, −16.23401193346523927460796487118, −15.0783330499773667964818472400, −14.70428811681281399505440544584, −13.616188949568667918278672051018, −12.83333441769104534874860377372, −12.118870983030035367890971879800, −11.31799464623606023036549239198, −10.88785142997416682022598419729, −9.76527334179094698436434578893, −8.915107453081902523612777904976, −8.01730687118513249373373174835, −7.50587463823421509435963887385, −5.74095244720620950253697580941, −5.10589478369619551933595629356, −4.22768063271560443930038208885, −3.48196840080215755767594039416, −2.39642258933176123622744933753, −1.22509367033133600366282103203, −0.47055326271997520690744456076, 0.929181075188073110622644078159, 2.42920841470881796452549906966, 3.74102960653495842349963062739, 4.34460285439653193591107080259, 5.17537263090236909422546288624, 6.28810904918994950533968276530, 7.10774905341696504737694070192, 7.72484582398498399474493390963, 8.622107534579506609056038162028, 9.1812446100085242776118440309, 10.69172029667400922322961706908, 11.23473498449522272269176851811, 12.251491977738767827262762897219, 12.996867399917266992524175691337, 14.13305438151365065515774958944, 14.696637020632583603899380713274, 15.16294745209348797791332769757, 16.1165332812714821204701560319, 16.825581567000507951222204057548, 17.60261305639090591744783644632, 18.474423425696640780097584414108, 18.99652214268750934911350259384, 19.96423337794038176337091809762, 21.09713895960939337223838801483, 21.71859699693893268237314740813

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