Properties

Degree 1
Conductor 89
Sign $0.261 - 0.965i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.142 + 0.989i)2-s + (0.877 + 0.479i)3-s + (−0.959 − 0.281i)4-s + (−0.909 − 0.415i)5-s + (−0.599 + 0.800i)6-s + (−0.936 + 0.349i)7-s + (0.415 − 0.909i)8-s + (0.540 + 0.841i)9-s + (0.540 − 0.841i)10-s + (−0.415 − 0.909i)11-s + (−0.707 − 0.707i)12-s + (0.479 − 0.877i)13-s + (−0.212 − 0.977i)14-s + (−0.599 − 0.800i)15-s + (0.841 + 0.540i)16-s + (−0.989 + 0.142i)17-s + ⋯
L(s,χ)  = 1  + (−0.142 + 0.989i)2-s + (0.877 + 0.479i)3-s + (−0.959 − 0.281i)4-s + (−0.909 − 0.415i)5-s + (−0.599 + 0.800i)6-s + (−0.936 + 0.349i)7-s + (0.415 − 0.909i)8-s + (0.540 + 0.841i)9-s + (0.540 − 0.841i)10-s + (−0.415 − 0.909i)11-s + (−0.707 − 0.707i)12-s + (0.479 − 0.877i)13-s + (−0.212 − 0.977i)14-s + (−0.599 − 0.800i)15-s + (0.841 + 0.540i)16-s + (−0.989 + 0.142i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.261 - 0.965i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.261 - 0.965i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(89\)
\( \varepsilon \)  =  $0.261 - 0.965i$
motivic weight  =  \(0\)
character  :  $\chi_{89} (51, \cdot )$
Sato-Tate  :  $\mu(88)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 89,\ (1:\ ),\ 0.261 - 0.965i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.3109127259 - 0.2378837008i$
$L(\frac12,\chi)$  $\approx$  $0.3109127259 - 0.2378837008i$
$L(\chi,1)$  $\approx$  0.6784291324 + 0.2725679191i
$L(1,\chi)$  $\approx$  0.6784291324 + 0.2725679191i

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

−30.63644117460093307386750545850, −29.51787673588679828812113677726, −28.61510395610342333420171354901, −27.25676195750420347360482751094, −26.293676977304498190560179067545, −25.70474687586980192662004262221, −23.817260306005224193195323407868, −23.051300775200739641286728880373, −21.88064687131659438954038030511, −20.25114740701125055552626567948, −20.00831370848357116998049023072, −18.774475765635651744047752253402, −18.2107140643722399483928655638, −16.32798172554591933496270003612, −14.951793524152059545423255007566, −13.73516943669985112518248745867, −12.75525431770410534036897183652, −11.75409679440076909251264872823, −10.295704696981648372766695043789, −9.2103119780474102909491701939, −7.945359552218968261060001362460, −6.81634876162091852009806627510, −4.186283511381597290394272594063, −3.32531101198418105674333056518, −1.89077710728346220144741992779, 0.16729157804480072773517404704, 3.17760423633864244535062895728, 4.35805950652656465265321797834, 5.86932934737004287039313882981, 7.5112497831815455814954033386, 8.506096377758708153241325446953, 9.29381439125479319815289170083, 10.7769214331834311867270945128, 12.858703004840605718267779211848, 13.62017772534262530539140914578, 15.208008271213317140526057062942, 15.753627101896409474040344416836, 16.47385593255364654975404554931, 18.248619981428601031159355884, 19.34653870349084470066904576780, 20.08998872933616719076211750339, 21.74865017512373341939227019904, 22.68592327950582512964369618379, 24.00580081834590216090999708359, 24.80649791624207608220853022663, 25.98588811037418543758053308065, 26.609503431178442807097267913368, 27.65268318078696282862203534875, 28.55977307400187396697019381477, 30.510989333736275713837091638284

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