Properties

Degree 1
Conductor 89
Sign $-0.514 + 0.857i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.654 + 0.755i)2-s + (−0.142 + 0.989i)3-s + (−0.142 − 0.989i)4-s + (0.841 + 0.540i)5-s + (−0.654 − 0.755i)6-s + (0.841 + 0.540i)7-s + (0.841 + 0.540i)8-s + (−0.959 − 0.281i)9-s + (−0.959 + 0.281i)10-s + (0.841 − 0.540i)11-s + 12-s + (−0.142 + 0.989i)13-s + (−0.959 + 0.281i)14-s + (−0.654 + 0.755i)15-s + (−0.959 + 0.281i)16-s + (−0.654 − 0.755i)17-s + ⋯
L(s,χ)  = 1  + (−0.654 + 0.755i)2-s + (−0.142 + 0.989i)3-s + (−0.142 − 0.989i)4-s + (0.841 + 0.540i)5-s + (−0.654 − 0.755i)6-s + (0.841 + 0.540i)7-s + (0.841 + 0.540i)8-s + (−0.959 − 0.281i)9-s + (−0.959 + 0.281i)10-s + (0.841 − 0.540i)11-s + 12-s + (−0.142 + 0.989i)13-s + (−0.959 + 0.281i)14-s + (−0.654 + 0.755i)15-s + (−0.959 + 0.281i)16-s + (−0.654 − 0.755i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.514 + 0.857i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.514 + 0.857i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(89\)
\( \varepsilon \)  =  $-0.514 + 0.857i$
motivic weight  =  \(0\)
character  :  $\chi_{89} (39, \cdot )$
Sato-Tate  :  $\mu(11)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 89,\ (0:\ ),\ -0.514 + 0.857i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.3951433559 + 0.6978543921i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.3951433559 + 0.6978543921i\)
\(L(\chi,1)\)  \(\approx\)  \(0.6400081919 + 0.5598040086i\)
\(L(1,\chi)\)  \(\approx\)  \(0.6400081919 + 0.5598040086i\)

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

−30.127422363828045085271367144428, −29.22018741242141992204026999558, −28.18412419858423829509506855430, −27.38422954358177506462372180007, −25.805277180374647418484910824891, −25.06885498086698112963905497989, −24.0199815639355367904679145810, −22.59958252405442202438538762909, −21.42263787205243070114696077125, −20.21577661426071455238829037878, −19.63642065915735715476383418846, −17.988237999505937750703132520664, −17.56957239437240121468648960311, −16.769344960973824628914540562396, −14.48742274077587775193504136833, −13.24031503012935915340613117088, −12.49377513716920543194864971735, −11.25832161119150261033615421790, −10.06998263371788748025492894407, −8.63116976299735651869682208101, −7.717316431084689291356166250787, −6.19911828284714245031632187741, −4.37789990188251279468999178588, −2.22110941692914854171145366191, −1.25331612305958951063505042462, 2.13624956024163401209057930668, 4.4527724706586027268345517501, 5.704029443126340233885692993289, 6.740241878374268276079306102590, 8.68216215638373646317487514158, 9.31552063556647749478886888877, 10.6186752454322957280072367682, 11.509994222789983003038931350494, 14.103467283321704421522161670216, 14.49281801232420803652178575181, 15.74218046042242202022705639565, 16.8660690054886048997148494041, 17.690120752625339585530291906908, 18.73557827629849685366316393967, 20.15403803970745553174837512253, 21.58063627690250508030638532584, 22.1458167746950067788754107736, 23.652390559436485296159479219462, 24.81863626757689901701826632022, 25.74604274494142372756488840610, 26.74207953760773639029943832622, 27.46877360977858570674873776872, 28.491572422337821761341260684206, 29.47332988694783693921840298540, 31.11829796609777664244250768633

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