Properties

Degree 1
Conductor 283
Sign $-0.815 - 0.579i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.860 − 0.509i)2-s + (−0.784 + 0.619i)3-s + (0.480 + 0.876i)4-s + (−0.231 − 0.972i)5-s + (0.991 − 0.133i)6-s + (−0.892 + 0.451i)7-s + (0.0334 − 0.999i)8-s + (0.231 − 0.972i)9-s + (−0.296 + 0.955i)10-s + (0.480 − 0.876i)11-s + (−0.920 − 0.390i)12-s + (0.991 − 0.133i)13-s + (0.997 + 0.0667i)14-s + (0.784 + 0.619i)15-s + (−0.538 + 0.842i)16-s + (0.997 − 0.0667i)17-s + ⋯
L(s,χ)  = 1  + (−0.860 − 0.509i)2-s + (−0.784 + 0.619i)3-s + (0.480 + 0.876i)4-s + (−0.231 − 0.972i)5-s + (0.991 − 0.133i)6-s + (−0.892 + 0.451i)7-s + (0.0334 − 0.999i)8-s + (0.231 − 0.972i)9-s + (−0.296 + 0.955i)10-s + (0.480 − 0.876i)11-s + (−0.920 − 0.390i)12-s + (0.991 − 0.133i)13-s + (0.997 + 0.0667i)14-s + (0.784 + 0.619i)15-s + (−0.538 + 0.842i)16-s + (0.997 − 0.0667i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.815 - 0.579i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.815 - 0.579i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(283\)
\( \varepsilon \)  =  $-0.815 - 0.579i$
motivic weight  =  \(0\)
character  :  $\chi_{283} (30, \cdot )$
Sato-Tate  :  $\mu(94)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 283,\ (1:\ ),\ -0.815 - 0.579i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.1327913893 - 0.4163302889i$
$L(\frac12,\chi)$  $\approx$  $0.1327913893 - 0.4163302889i$
$L(\chi,1)$  $\approx$  0.4792726177 - 0.1348869372i
$L(1,\chi)$  $\approx$  0.4792726177 - 0.1348869372i

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

−25.690419290035094251671165834228, −25.103304536976984665022997132271, −23.77907841555839420656322898524, −22.95599237154495825906541683752, −22.788865541549919496369324677222, −21.17123726986053953333449872028, −19.61618620543888213989586683709, −19.26205990166008420908909233078, −18.25416366575648239689828330415, −17.640860429606924053435104533896, −16.59159407510327590315845204283, −15.92301609106295175809507082191, −14.75406427238313343450520359312, −13.79196761830738510374853314445, −12.48278954808414740614087185061, −11.44841138228821784454662635726, −10.459276286401106692093760678512, −9.91101527019559898655367672294, −8.314945445039336897371399914540, −7.29093820343346672913380984861, −6.52110194853302025472814368384, −6.02328835175548870146438329792, −4.233810779530517393892500233767, −2.50376115663121251004250173153, −1.0935322752645216990582322728, 0.25482409974443452409172238527, 1.27685565784306585284539452987, 3.27455077551441551553308859073, 4.070461948191391907439924120196, 5.68157516084385898612233320044, 6.48048575676896495482308637664, 8.174272003904186665514065151451, 8.98122432561486745188575787703, 9.7886339314900688912124677115, 10.78416542094358684558057897925, 11.8396335690548205599086327859, 12.37639588630981925503245464386, 13.461076598711831183871693726660, 15.412382641053829745252119292119, 16.15966826474753933435693799807, 16.646851596351499817436226669168, 17.516495690184048481392532010950, 18.71464874503363016951114663459, 19.4155964570103766345345692082, 20.54186469567792088709501910858, 21.27627013464634319938562342682, 21.99511507157346445539925213783, 23.101306360333785633744376422112, 24.07393092941026410786564170319, 25.30936198738626157271777886781

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