Properties

Degree 1
Conductor 89
Sign $-0.959 - 0.280i$
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 + (0.599 + 0.800i)3-s + (−0.142 − 0.989i)4-s + (−0.540 + 0.841i)5-s + (−0.997 − 0.0713i)6-s + (0.977 − 0.212i)7-s + (0.841 + 0.540i)8-s + (−0.281 + 0.959i)9-s + (−0.281 − 0.959i)10-s + (−0.841 + 0.540i)11-s + (0.707 − 0.707i)12-s + (−0.800 + 0.599i)13-s + (−0.479 + 0.877i)14-s + (−0.997 + 0.0713i)15-s + (−0.959 + 0.281i)16-s + (0.755 − 0.654i)17-s + ⋯
L(s,χ)  = 1  + (−0.654 + 0.755i)2-s + (0.599 + 0.800i)3-s + (−0.142 − 0.989i)4-s + (−0.540 + 0.841i)5-s + (−0.997 − 0.0713i)6-s + (0.977 − 0.212i)7-s + (0.841 + 0.540i)8-s + (−0.281 + 0.959i)9-s + (−0.281 − 0.959i)10-s + (−0.841 + 0.540i)11-s + (0.707 − 0.707i)12-s + (−0.800 + 0.599i)13-s + (−0.479 + 0.877i)14-s + (−0.997 + 0.0713i)15-s + (−0.959 + 0.281i)16-s + (0.755 − 0.654i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.959 - 0.280i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.959 - 0.280i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(89\)
\( \varepsilon \)  =  $-0.959 - 0.280i$
motivic weight  =  \(0\)
character  :  $\chi_{89} (66, \cdot )$
Sato-Tate  :  $\mu(88)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 89,\ (1:\ ),\ -0.959 - 0.280i)$
$L(\chi,\frac{1}{2})$  $\approx$  $-0.1381042602 + 0.9665666142i$
$L(\frac12,\chi)$  $\approx$  $-0.1381042602 + 0.9665666142i$
$L(\chi,1)$  $\approx$  0.5297934749 + 0.6214146852i
$L(1,\chi)$  $\approx$  0.5297934749 + 0.6214146852i

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

−29.71149718426686459135854807948, −28.65356798274721326448121454245, −27.66776879886930904289761081027, −26.71332113351331436258502173821, −25.57078953863434558836835254387, −24.37539207004028342142240566562, −23.723319815931840429429081729753, −21.849773140902656839066447810649, −20.671165251351062667874449716856, −20.06891329955565945792472431991, −18.99599628032726691260334552728, −18.04001570657869764609342183000, −17.04839056182452621129887694766, −15.56877795081218737102695162506, −14.03339296657347511993461173052, −12.70468339125195999278455246217, −12.10545325870974238846370807408, −10.767071461022746685798014732480, −9.10189748298812912410796192203, −8.14494611720618092861810173264, −7.546114762194576898377959075440, −5.11290809623813935211653252459, −3.363784757764492583205427776698, −1.904507296779007863823671432196, −0.51223301305546275628272008598, 2.238876019834305171651074453433, 4.18312181160717908135467974150, 5.43239339524486452905781236191, 7.53705867467785100268052365706, 7.84580023676842731257211131778, 9.55431371749941681937473598383, 10.39905427613841585136117208013, 11.5600859910075365167046145339, 14.04506207268507869758242890825, 14.5990536192365561960950068903, 15.54640544238151303142326260666, 16.55106934769875609739380844226, 17.92213259714232660603467554989, 18.88830154609539001797573302380, 20.01274219766920466766374094390, 21.07433671461878428479655531258, 22.525126525182963854810140302028, 23.54156272274521811535898788614, 24.71857922942493007711240908819, 25.89776221459079896372875556396, 26.65767836251998234162342913323, 27.30535700213150186909456758581, 28.21787239423755579289351698860, 29.8050578422993255135757530194, 31.2801612373556948540007230328

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