Properties

Degree $1$
Conductor $283$
Sign $0.790 - 0.611i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.741 − 0.670i)2-s + (0.991 − 0.133i)3-s + (0.100 + 0.994i)4-s + (0.964 + 0.264i)5-s + (−0.824 − 0.565i)6-s + (0.860 + 0.509i)7-s + (0.593 − 0.805i)8-s + (0.964 − 0.264i)9-s + (−0.538 − 0.842i)10-s + (0.100 − 0.994i)11-s + (0.231 + 0.972i)12-s + (−0.824 − 0.565i)13-s + (−0.296 − 0.955i)14-s + (0.991 + 0.133i)15-s + (−0.979 + 0.199i)16-s + (−0.296 + 0.955i)17-s + ⋯
L(s,χ)  = 1  + (−0.741 − 0.670i)2-s + (0.991 − 0.133i)3-s + (0.100 + 0.994i)4-s + (0.964 + 0.264i)5-s + (−0.824 − 0.565i)6-s + (0.860 + 0.509i)7-s + (0.593 − 0.805i)8-s + (0.964 − 0.264i)9-s + (−0.538 − 0.842i)10-s + (0.100 − 0.994i)11-s + (0.231 + 0.972i)12-s + (−0.824 − 0.565i)13-s + (−0.296 − 0.955i)14-s + (0.991 + 0.133i)15-s + (−0.979 + 0.199i)16-s + (−0.296 + 0.955i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.790 - 0.611i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.790 - 0.611i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(283\)
Sign: $0.790 - 0.611i$
Motivic weight: \(0\)
Character: $\chi_{283} (158, \cdot )$
Sato-Tate group: $\mu(47)$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 283,\ (0:\ ),\ 0.790 - 0.611i)\)

Particular Values

\(L(\chi,\frac{1}{2})\) \(\approx\) \(1.431529161 - 0.4890982698i\)
\(L(\frac12,\chi)\) \(\approx\) \(1.431529161 - 0.4890982698i\)
\(L(\chi,1)\) \(\approx\) \(1.210126815 - 0.3160645582i\)
\(L(1,\chi)\) \(\approx\) \(1.210126815 - 0.3160645582i\)

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

−25.69289156905257548455581752385, −24.8612292202536757053008873343, −24.3872216915660383583338113159, −23.27930382609289286771625701839, −21.91493436131518188117273850366, −20.661671564676134415640793150508, −20.4221675594557970051688081482, −19.18808333796259762141730692292, −18.2961845438691171970520632242, −17.30521455405144235247887026018, −16.77229246579548250829505422579, −15.3595268225984573543271669867, −14.61916254596347651904425514394, −13.974713527020016099956126309427, −12.99418113164472785532333910214, −11.3209917244869102927682684077, −9.95707171923918386741348841289, −9.60219649631859884491796341880, −8.56688843076241124327695368198, −7.52069081879558036681406727520, −6.80384060177935614616273801546, −5.17934640734468869373068643988, −4.396500406852027162502523759612, −2.24493625666192782048444779743, −1.60101903951695164555180487277, 1.4832742421598215353864895878, 2.372999657836009176368306207132, 3.24294738658480294155047329566, 4.7802853760276468866507195032, 6.39332098926158278959831528970, 7.635392544360212927948518989954, 8.64784013694368124008048159083, 9.13201883977731500964354643298, 10.385640966263232913842391627922, 11.05905911873331391345894124744, 12.57626981571972916136020265083, 13.18888136162101804572336694994, 14.39754471729589365094298350881, 15.07953351029783794591628791705, 16.57804198339138524814907997524, 17.58978027462479524759048643633, 18.253748549077434848700555771260, 19.17498397781068803149794435596, 19.886548376471948394676258997, 21.12201504254518089163632782480, 21.41282764539081108525750744634, 22.22019471106626595008992445112, 24.173735618374999861144761294105, 24.83330944215474802062816053942, 25.57218287078225787590602908281

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