Properties

Degree $1$
Conductor $283$
Sign $-0.930 - 0.365i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.100 + 0.994i)2-s + (−0.964 − 0.264i)3-s + (−0.979 − 0.199i)4-s + (−0.860 + 0.509i)5-s + (0.359 − 0.933i)6-s + (0.480 − 0.876i)7-s + (0.296 − 0.955i)8-s + (0.860 + 0.509i)9-s + (−0.420 − 0.907i)10-s + (−0.979 + 0.199i)11-s + (0.892 + 0.451i)12-s + (0.359 − 0.933i)13-s + (0.824 + 0.565i)14-s + (0.964 − 0.264i)15-s + (0.920 + 0.390i)16-s + (0.824 − 0.565i)17-s + ⋯
L(s,χ)  = 1  + (−0.100 + 0.994i)2-s + (−0.964 − 0.264i)3-s + (−0.979 − 0.199i)4-s + (−0.860 + 0.509i)5-s + (0.359 − 0.933i)6-s + (0.480 − 0.876i)7-s + (0.296 − 0.955i)8-s + (0.860 + 0.509i)9-s + (−0.420 − 0.907i)10-s + (−0.979 + 0.199i)11-s + (0.892 + 0.451i)12-s + (0.359 − 0.933i)13-s + (0.824 + 0.565i)14-s + (0.964 − 0.264i)15-s + (0.920 + 0.390i)16-s + (0.824 − 0.565i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.930 - 0.365i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.930 - 0.365i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(283\)
Sign: $-0.930 - 0.365i$
Motivic weight: \(0\)
Character: $\chi_{283} (33, \cdot )$
Sato-Tate group: $\mu(94)$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 283,\ (1:\ ),\ -0.930 - 0.365i)\)

Particular Values

\(L(\chi,\frac{1}{2})\) \(\approx\) \(0.02802979097 + 0.1478637868i\)
\(L(\frac12,\chi)\) \(\approx\) \(0.02802979097 + 0.1478637868i\)
\(L(\chi,1)\) \(\approx\) \(0.5029483587 + 0.1923089973i\)
\(L(1,\chi)\) \(\approx\) \(0.5029483587 + 0.1923089973i\)

Euler product

   \(L(\chi,s) = \displaystyle\prod_p (1- \chi(p) p^{-s})^{-1}\)
   \(L(s,\chi) = \displaystyle\prod_p (1- \chi(p) p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−24.502598813540191974667935072, −23.53499439806035698041878840411, −23.179211894626422987219610555170, −21.810018019023610392365331335372, −21.31983138807732045599114413509, −20.560448990824241931138225989273, −19.16306144584097655875102632159, −18.64522826800908340219615898285, −17.68677533754114344556477486024, −16.656044737583554464110232429078, −15.77181272580969392863786941279, −14.73744375654267298472256916085, −13.1794034367325428041008916902, −12.36932393407207519025808312264, −11.61814721666819424644316458448, −11.01300333039470066693502705152, −9.89798380141013756498111315850, −8.71787580769284960172855053875, −7.93764927076727515371472299663, −6.16169434627334415472667761927, −4.8745322771825141510544125364, −4.41492967900447001690712501362, −2.901589350041646856012485279701, −1.35481561468474297263564914242, −0.0724665076941418496812161912, 1.05713206816733147714053789021, 3.492115549600735751633123924513, 4.67760108451826060215462974235, 5.54931544005280686606413277627, 6.73178902543231413070505435365, 7.71683253414211011853526155313, 8.00830651086364336274738485179, 10.114796268462754064847705542682, 10.59627309090948676314529803099, 11.8369792018236855715414078386, 12.92685094615872758446089818618, 13.87775141169965583142290633501, 15.03178452884481984308193270296, 15.85463299634480873260939745725, 16.613337535331275155288164829, 17.61493030452415432579200341523, 18.30483576231906707656445743616, 19.04294484653529225587253635863, 20.383860641702491693447332767297, 21.65532254585787781995329585587, 22.870973771431543178635860117632, 23.305890084697981988105347301146, 23.67514375350114723789435222525, 24.863502542040656538488593589407, 25.82187528819827051076704877529

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