Properties

Degree 1
Conductor 89
Sign $-0.314 + 0.949i$
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

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.314 + 0.949i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.314 + 0.949i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(89\)
\( \varepsilon \)  =  $-0.314 + 0.949i$
motivic weight  =  \(0\)
character  :  $\chi_{89} (38, \cdot )$
Sato-Tate  :  $\mu(88)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 89,\ (1:\ ),\ -0.314 + 0.949i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.3622480504 + 0.5016796543i$
$L(\frac12,\chi)$  $\approx$  $0.3622480504 + 0.5016796543i$
$L(\chi,1)$  $\approx$  0.5659105210 + 0.2058649345i
$L(1,\chi)$  $\approx$  0.5659105210 + 0.2058649345i

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

−30.00160573574821803512063179038, −28.64337507607385184136386828330, −27.93759231806812940130962148134, −27.14959918026148722664857857955, −26.36499251017509872781280630107, −24.34935573573603653671363008350, −23.08562991664038453301946814804, −22.54160781200822592007874665420, −21.375318960173613116061670639493, −20.43351792239674254968555553278, −19.25772381055658863637020158910, −17.85749558322061626013851116487, −17.52481776261290012199689362955, −15.6097047429511724502779521994, −14.82948976512577306628657103916, −12.89808562464358568850017013684, −11.84978640159020520953884835277, −11.080905511876030498783017520986, −10.182108570441094051585991504240, −8.65179188573882496309541689039, −7.23393304302823320620949914933, −5.07249289417163253010005646264, −4.3240482332603530419563868950, −2.570574099345257592197792613661, −0.42450247693666845046464310423, 1.10078580535501323015628029375, 4.3019863533095954167971147661, 5.19439994126594826895034694221, 6.67886724097108587974527829579, 7.740848921107000607610226785339, 8.67352743171861177313363765693, 10.6480660339018926114460675301, 11.71651315190584091946560434287, 13.036450163820425324935038517116, 14.225393219256714734461698110755, 15.60433160176613266673056876150, 16.59617045011522926578612320712, 17.323957277088161678089620896905, 18.57127964401670474145140196912, 19.38206235525908973034244949630, 21.20866987031282066972802725420, 22.49877743740774700709745173327, 23.66995470997430826829221873219, 24.00864639957128242816803411639, 24.870704295354126599994256026252, 26.73250473542954988846661802525, 27.19620020911243276916959052359, 28.27699141177248176210514048915, 29.35415651043178122202651181869, 30.92625477962938825710407317490

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