Properties

Degree 1
Conductor 89
Sign $-0.182 + 0.983i$
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 + (0.800 + 0.599i)3-s + (−0.142 + 0.989i)4-s + (0.540 + 0.841i)5-s + (−0.0713 − 0.997i)6-s + (−0.212 + 0.977i)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 + (−0.599 + 0.800i)13-s + (0.877 − 0.479i)14-s + (−0.0713 + 0.997i)15-s + (−0.959 − 0.281i)16-s + (−0.755 − 0.654i)17-s + ⋯
L(s,χ)  = 1  + (−0.654 − 0.755i)2-s + (0.800 + 0.599i)3-s + (−0.142 + 0.989i)4-s + (0.540 + 0.841i)5-s + (−0.0713 − 0.997i)6-s + (−0.212 + 0.977i)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 + (−0.599 + 0.800i)13-s + (0.877 − 0.479i)14-s + (−0.0713 + 0.997i)15-s + (−0.959 − 0.281i)16-s + (−0.755 − 0.654i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.182 + 0.983i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.182 + 0.983i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(89\)
\( \varepsilon \)  =  $-0.182 + 0.983i$
motivic weight  =  \(0\)
character  :  $\chi_{89} (14, \cdot )$
Sato-Tate  :  $\mu(88)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 89,\ (1:\ ),\ -0.182 + 0.983i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.8035023608 + 0.9660466464i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.8035023608 + 0.9660466464i\)
\(L(\chi,1)\)  \(\approx\)  \(0.9148053658 + 0.2780762479i\)
\(L(1,\chi)\)  \(\approx\)  \(0.9148053658 + 0.2780762479i\)

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

−29.66942115219438562922843619232, −29.010180325087663929581901506222, −27.695144840037103799696062206294, −26.50182679973792978772256605666, −25.704105524793282767674583354824, −24.86303148959643885429632853182, −23.935177264429123217806149224362, −23.09254377237762692188060589823, −21.00940902488127564659544805442, −20.031981237648302872729136075120, −19.30608262990898058486319706340, −17.74118975136775136219014226507, −17.27850603937110849983748446927, −15.81285231768764980111370864688, −14.71183138259170256048274305883, −13.45488733197152854945482883880, −12.79073328926913945094833034169, −10.433820979820747718328110304088, −9.506865278147114679650880787547, −8.24893804505182216261071779820, −7.42543414458771602209818969114, −6.07076147473105896187849635021, −4.47055827734584369235429098984, −2.12447549035521357354775518880, −0.62755133181194289322458466678, 2.40355637160477953677864645242, 2.830564871206753907732120396419, 4.69883334568798661375774321044, 6.79192552809507817670030348404, 8.39085934183671507471711460869, 9.2791180083174310178241827152, 10.30719812210233918825334928841, 11.304486873877405721125837519488, 12.91380376169940591278784575652, 14.03780398013912990744513762083, 15.32703633007801363468260106552, 16.451523613850737211636325788795, 17.99623754356996427037171185433, 18.86859772354948488037246410355, 19.654016850652107870849315386565, 21.24537615605069912796660349045, 21.523996307087088130560817482413, 22.55519049192061944643047531434, 24.7192592008210579555214951175, 25.69804914285747585818444029834, 26.46306276687793545210069242654, 27.19343227216274349525933915900, 28.590723897978487685514105806557, 29.29313061103912045068209427229, 30.68552790821105923265129854375

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