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: $\chi_{283} (122, \cdot )$
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

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

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