Properties

Degree $1$
Conductor $283$
Sign $-0.429 - 0.903i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.695 + 0.718i)2-s + (0.538 + 0.842i)3-s + (−0.0334 − 0.999i)4-s + (0.420 + 0.907i)5-s + (−0.979 − 0.199i)6-s + (−0.645 + 0.763i)7-s + (0.741 + 0.670i)8-s + (−0.420 + 0.907i)9-s + (−0.944 − 0.328i)10-s + (−0.0334 + 0.999i)11-s + (0.824 − 0.565i)12-s + (−0.979 − 0.199i)13-s + (−0.100 − 0.994i)14-s + (−0.538 + 0.842i)15-s + (−0.997 + 0.0667i)16-s + (−0.100 + 0.994i)17-s + ⋯
L(s,χ)  = 1  + (−0.695 + 0.718i)2-s + (0.538 + 0.842i)3-s + (−0.0334 − 0.999i)4-s + (0.420 + 0.907i)5-s + (−0.979 − 0.199i)6-s + (−0.645 + 0.763i)7-s + (0.741 + 0.670i)8-s + (−0.420 + 0.907i)9-s + (−0.944 − 0.328i)10-s + (−0.0334 + 0.999i)11-s + (0.824 − 0.565i)12-s + (−0.979 − 0.199i)13-s + (−0.100 − 0.994i)14-s + (−0.538 + 0.842i)15-s + (−0.997 + 0.0667i)16-s + (−0.100 + 0.994i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.429 - 0.903i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.429 - 0.903i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(283\)
Sign: $-0.429 - 0.903i$
Motivic weight: \(0\)
Character: $\chi_{283} (58, \cdot )$
Sato-Tate group: $\mu(94)$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 283,\ (1:\ ),\ -0.429 - 0.903i)\)

Particular Values

\(L(\chi,\frac{1}{2})\) \(\approx\) \(-0.6027707499 + 0.9538798421i\)
\(L(\frac12,\chi)\) \(\approx\) \(-0.6027707499 + 0.9538798421i\)
\(L(\chi,1)\) \(\approx\) \(0.4400152462 + 0.7188517138i\)
\(L(1,\chi)\) \(\approx\) \(0.4400152462 + 0.7188517138i\)

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

−24.833355981195081044502064475015, −24.158497276375796957438227748357, −22.904653001350589550332376215324, −21.750925914529500937987859514514, −20.753702819559980450677527710949, −20.02631829525606128174414621622, −19.42473077763621591153839473841, −18.52763259196656122172771681896, −17.46295397015266412207828368827, −16.79742798096948683953108494916, −15.86290192718025025893326364012, −13.782861496959308139029100892463, −13.64533902060202154723576926790, −12.46807587945913207461961728450, −11.81077163556741257583210284886, −10.39898219482820730989467597618, −9.30375225666765775153966272987, −8.79634978084904900852919856683, −7.537820073963375047603943570327, −6.81142358836761366480718426451, −5.10453838912333480888755388285, −3.5190223146992203834598157000, −2.606109167418416195216839128318, −1.174846587807873413093595160849, −0.436342422898845606144600318275, 2.06277557937155880756626070319, 3.01501511907073385989258060370, 4.69561762232175967646359803469, 5.7500103094180832525702585779, 6.831385616681818143183132059173, 7.86192594899353163896640764127, 9.02651093784371890578695661866, 10.00135057363322891761009546030, 10.17453124078595970825867714137, 11.6788864409050497364319848654, 13.269005715807695337077322042820, 14.409797244718033927754607067, 15.17378839913311825409298281363, 15.50285580315283378492935433168, 16.85603377642277029949208756194, 17.58624186418008115562484216886, 18.754049882941785755112478316277, 19.378806712350284348355968818221, 20.340593619569320076182814799719, 21.587417979993199350787096615832, 22.44786550384200313260497655318, 23.03292780564319806082024681914, 24.81978096744031310453189404379, 25.10664621716583728287362396099, 26.26651083335681487559953066989

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