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Properties

Label 1-189-189.88-r0-0-0
Degree $1$
Conductor $189$
Sign $0.453 + 0.891i$
Analytic cond. $0.877712$
Root an. cond. $0.877712$
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  + (0.173 + 0.984i)2-s + (−0.939 + 0.342i)4-s + (0.173 − 0.984i)5-s + (−0.5 − 0.866i)8-s + 10-s + (0.173 + 0.984i)11-s + (0.766 + 0.642i)13-s + (0.766 − 0.642i)16-s + 17-s + 19-s + (0.173 + 0.984i)20-s + (−0.939 + 0.342i)22-s + (0.766 + 0.642i)23-s + (−0.939 − 0.342i)25-s + (−0.5 + 0.866i)26-s + ⋯
L(s)  = 1  + (0.173 + 0.984i)2-s + (−0.939 + 0.342i)4-s + (0.173 − 0.984i)5-s + (−0.5 − 0.866i)8-s + 10-s + (0.173 + 0.984i)11-s + (0.766 + 0.642i)13-s + (0.766 − 0.642i)16-s + 17-s + 19-s + (0.173 + 0.984i)20-s + (−0.939 + 0.342i)22-s + (0.766 + 0.642i)23-s + (−0.939 − 0.342i)25-s + (−0.5 + 0.866i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.453 + 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.453 + 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.453 + 0.891i$
Analytic conductor: \(0.877712\)
Root analytic conductor: \(0.877712\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (88, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 189,\ (0:\ ),\ 0.453 + 0.891i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.044313639 + 0.6404017618i\)
\(L(\frac12)\) \(\approx\) \(1.044313639 + 0.6404017618i\)
\(L(1)\) \(\approx\) \(1.034676102 + 0.4542708951i\)
\(L(1)\) \(\approx\) \(1.034676102 + 0.4542708951i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.173 + 0.984i)T \)
5 \( 1 + (0.173 - 0.984i)T \)
11 \( 1 + (0.173 + 0.984i)T \)
13 \( 1 + (0.766 + 0.642i)T \)
17 \( 1 + T \)
19 \( 1 + T \)
23 \( 1 + (0.766 + 0.642i)T \)
29 \( 1 + (0.766 - 0.642i)T \)
31 \( 1 + (-0.939 + 0.342i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (0.766 + 0.642i)T \)
43 \( 1 + (-0.939 - 0.342i)T \)
47 \( 1 + (-0.939 - 0.342i)T \)
53 \( 1 + (-0.5 - 0.866i)T \)
59 \( 1 + (0.766 + 0.642i)T \)
61 \( 1 + (-0.939 - 0.342i)T \)
67 \( 1 + (0.173 - 0.984i)T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 + (-0.5 + 0.866i)T \)
79 \( 1 + (0.173 + 0.984i)T \)
83 \( 1 + (0.766 - 0.642i)T \)
89 \( 1 + T \)
97 \( 1 + (-0.939 - 0.342i)T \)
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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

−27.09740286639238981218367965805, −26.28284042423645566126904226726, −25.16032048144191883082653340300, −23.79460631939115505319921637849, −22.86259795162366415429409277539, −22.15219457979153816024800298280, −21.25627060125515658682399698254, −20.34860836506480884099362553299, −19.14302027283227381127004243787, −18.53767591155783156050411907931, −17.69224982466146614723488341908, −16.28649272133739805631738587155, −14.87062981313489570706145920056, −14.05818496840966066889132239003, −13.20824662269498781522422250458, −11.92711418426406851668405389903, −10.970821476390356864009110726209, −10.293970116458570874345443747347, −9.08828973632214830387639436542, −7.873758347810128847824594814957, −6.273084369791884767664262355217, −5.247447153546572459138892462383, −3.49601573319456912630969288241, −2.96215345481123282547604160626, −1.226369930478038293095792033335, 1.3629583064068997384659541506, 3.61489381314884584537595232401, 4.786056771248106622576435267622, 5.650725332861050933232344236792, 6.953474845399503450293077541557, 8.00299808447714809790456010520, 9.11132479808128202915322965249, 9.83536391796834131801970685009, 11.765798801006580745791906487776, 12.71826220069119484254895937682, 13.63828494140203870565655185869, 14.60930554197102354735341840200, 15.80390980201954526003312682630, 16.4843241979071345450009967490, 17.433670077283846172357244520255, 18.28453687796944918398272787395, 19.57811813524147007551731550346, 20.82987657954113622418174068757, 21.55045266733488437300631739205, 22.9571591307116625611684984374, 23.47604442901040182096797190566, 24.587297594253430050517388621565, 25.27604876612926350404598151878, 26.0590022361746112598148845488, 27.27671612011998391940805961587

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