Properties

Degree 1
Conductor 283
Sign $-0.0427 + 0.999i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.784 − 0.619i)2-s + (−0.741 + 0.670i)3-s + (0.231 − 0.972i)4-s + (0.100 + 0.994i)5-s + (−0.166 + 0.986i)6-s + (−0.979 + 0.199i)7-s + (−0.420 − 0.907i)8-s + (0.100 − 0.994i)9-s + (0.695 + 0.718i)10-s + (0.231 + 0.972i)11-s + (0.480 + 0.876i)12-s + (−0.166 + 0.986i)13-s + (−0.645 + 0.763i)14-s + (−0.741 − 0.670i)15-s + (−0.892 − 0.451i)16-s + (−0.645 − 0.763i)17-s + ⋯
L(s,χ)  = 1  + (0.784 − 0.619i)2-s + (−0.741 + 0.670i)3-s + (0.231 − 0.972i)4-s + (0.100 + 0.994i)5-s + (−0.166 + 0.986i)6-s + (−0.979 + 0.199i)7-s + (−0.420 − 0.907i)8-s + (0.100 − 0.994i)9-s + (0.695 + 0.718i)10-s + (0.231 + 0.972i)11-s + (0.480 + 0.876i)12-s + (−0.166 + 0.986i)13-s + (−0.645 + 0.763i)14-s + (−0.741 − 0.670i)15-s + (−0.892 − 0.451i)16-s + (−0.645 − 0.763i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.0427 + 0.999i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.0427 + 0.999i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(283\)
\( \varepsilon \)  =  $-0.0427 + 0.999i$
motivic weight  =  \(0\)
character  :  $\chi_{283} (64, \cdot )$
Sato-Tate  :  $\mu(47)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 283,\ (0:\ ),\ -0.0427 + 0.999i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.6543093178 + 0.6828957163i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.6543093178 + 0.6828957163i\)
\(L(\chi,1)\)  \(\approx\)  \(0.9906126539 + 0.1774007776i\)
\(L(1,\chi)\)  \(\approx\)  \(0.9906126539 + 0.1774007776i\)

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

−25.02126403532466108507045564144, −24.170807972289081327967569817907, −23.8801544684703849876293809322, −22.450596828826837034177680583138, −22.32204367385844098287621443294, −21.01550364224491552328712909969, −19.91054103481632544869187358730, −19.02928534331596714083419886684, −17.44744180750711759975276144735, −17.12942189920711734571174289777, −16.12793612332941254464658647258, −15.48926147665488338394738886694, −13.80507402337197875590311634343, −13.15763105120671073123022489362, −12.60589038994700463916025787772, −11.65706993198172952777512164403, −10.455873856282302042789914554441, −8.80754353680452043801875821132, −7.97887533962871048651820287792, −6.67644894130940703826458005779, −6.012379775374303353906325542879, −5.09595015177228473161739434817, −3.93055157577407097224557615003, −2.46074173243110973334869570283, −0.5096170384093255626593527199, 1.9496458065604988372981958499, 3.270520750175111644487349431622, 4.11569704480772197445846612945, 5.28878446984017970888567173605, 6.45556581726432198472495541433, 6.91809233879713938615455445115, 9.52128434321037728329282566969, 9.79476165469158747041025312843, 10.90389377312752702050769724651, 11.7562173524937808842462195749, 12.5020223314056973823128456235, 13.78291469602184390734327717653, 14.72719017700495670620284522024, 15.532976592882074592511864995321, 16.34877747665383135986664800839, 17.7197659303824255374587244516, 18.638606914472675883338845214677, 19.53816629143327238238930799058, 20.58808328575654861967144402959, 21.640843955003998285181768197656, 22.17339955974961084648400534037, 22.96563668311965419143819087124, 23.362390973631216099859389299101, 24.822507715228783072566109237295, 25.89889148239803217805054666807

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