Properties

Degree 1
Conductor $ 2^{3} \cdot 11 $
Sign $-0.286 + 0.958i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.809 + 0.587i)3-s + (−0.309 − 0.951i)5-s + (0.809 + 0.587i)7-s + (0.309 − 0.951i)9-s + (−0.309 + 0.951i)13-s + (0.809 + 0.587i)15-s + (0.309 + 0.951i)17-s + (−0.809 + 0.587i)19-s − 21-s − 23-s + (−0.809 + 0.587i)25-s + (0.309 + 0.951i)27-s + (0.809 + 0.587i)29-s + (−0.309 + 0.951i)31-s + (0.309 − 0.951i)35-s + ⋯
L(s,χ)  = 1  + (−0.809 + 0.587i)3-s + (−0.309 − 0.951i)5-s + (0.809 + 0.587i)7-s + (0.309 − 0.951i)9-s + (−0.309 + 0.951i)13-s + (0.809 + 0.587i)15-s + (0.309 + 0.951i)17-s + (−0.809 + 0.587i)19-s − 21-s − 23-s + (−0.809 + 0.587i)25-s + (0.309 + 0.951i)27-s + (0.809 + 0.587i)29-s + (−0.309 + 0.951i)31-s + (0.309 − 0.951i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.286 + 0.958i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 88 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.286 + 0.958i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(88\)    =    \(2^{3} \cdot 11\)
\( \varepsilon \)  =  $-0.286 + 0.958i$
motivic weight  =  \(0\)
character  :  $\chi_{88} (3, \cdot )$
Sato-Tate  :  $\mu(10)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 88,\ (1:\ ),\ -0.286 + 0.958i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.5379672705 + 0.7222362654i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.5379672705 + 0.7222362654i\)
\(L(\chi,1)\)  \(\approx\)  \(0.7544201683 + 0.2254452602i\)
\(L(1,\chi)\)  \(\approx\)  \(0.7544201683 + 0.2254452602i\)

Euler product

\[\begin{aligned}L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]
\[\begin{aligned}L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−30.157471266760955339019509104429, −29.206587494723186983995216007898, −27.70655267128661182189185200892, −27.16185103480194576044785034307, −25.76318759561328284788759482390, −24.51272350827212491144800694670, −23.51900252688907124016888526589, −22.73383990359546690981187964629, −21.75031347072144199337214722873, −20.23044707444550582907302132238, −19.02804251005549747228003278223, −17.9779632035732095276997816909, −17.31049473321381449245979693407, −15.85561384648880259102292210610, −14.555518108344083399394838360463, −13.457200432559701110977975652667, −12.01852821742871836922482544763, −11.073631809210209041022904154547, −10.20228397442539215374316570459, −7.93883490845675378442245332401, −7.188318316037918327520181603262, −5.84619498761786799930468828347, −4.375267203339177563519492033082, −2.42006020731366506099652799193, −0.48039704911832046141342113778, 1.54908070691974308815909262611, 4.07661181760113763867905255468, 5.00204012520231684109689979898, 6.21846396710851520121857741438, 8.12380395916241158045170991383, 9.20658259786213579189227322050, 10.60501969387131997435116337733, 11.88220812089565766902717285000, 12.47477288389112366776316117739, 14.38070898743556597492460844903, 15.5239456843187996220246169586, 16.57875295019115192410832754313, 17.37087588201107213450517135859, 18.64727375899107117811179121548, 20.09501667301295572888085489169, 21.31945331335786340483824991135, 21.77498525093982904737272076387, 23.516125664946322230304971862492, 23.92889141244392233077568412562, 25.24645847054428223901972314260, 26.7587561200385200324488019450, 27.7060162870340339058512820666, 28.32242011369709066519646591276, 29.27058989329998240294421071537, 30.749049288467832805811965833300

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