L-function 1-189-189.88-r0-0-0

• 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: $0$

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 + ⋯

Functional equation

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

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

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

Euler product

   $L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$

	$p$	$F_p(T)$
bad	3	$1$
	7	$1$
good	2	$1 + (0.173 + 0.984i)T$
	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	$1 + (-0.939 + 0.342i)T$

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