Properties

Degree 1
Conductor 293
Sign $-0.681 - 0.732i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.744 + 0.668i)2-s + (0.192 + 0.981i)3-s + (0.107 − 0.994i)4-s + (−0.618 + 0.785i)5-s + (−0.798 − 0.601i)6-s + (−0.317 − 0.948i)7-s + (0.584 + 0.811i)8-s + (−0.925 + 0.377i)9-s + (−0.0645 − 0.997i)10-s + (0.772 + 0.635i)11-s + (0.996 − 0.0859i)12-s + (0.276 + 0.961i)13-s + (0.869 + 0.493i)14-s + (−0.890 − 0.455i)15-s + (−0.976 − 0.213i)16-s + (−0.744 + 0.668i)17-s + ⋯
L(s,χ)  = 1  + (−0.744 + 0.668i)2-s + (0.192 + 0.981i)3-s + (0.107 − 0.994i)4-s + (−0.618 + 0.785i)5-s + (−0.798 − 0.601i)6-s + (−0.317 − 0.948i)7-s + (0.584 + 0.811i)8-s + (−0.925 + 0.377i)9-s + (−0.0645 − 0.997i)10-s + (0.772 + 0.635i)11-s + (0.996 − 0.0859i)12-s + (0.276 + 0.961i)13-s + (0.869 + 0.493i)14-s + (−0.890 − 0.455i)15-s + (−0.976 − 0.213i)16-s + (−0.744 + 0.668i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 293 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.681 - 0.732i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 293 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.681 - 0.732i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(293\)
\( \varepsilon \)  =  $-0.681 - 0.732i$
motivic weight  =  \(0\)
character  :  $\chi_{293} (38, \cdot )$
Sato-Tate  :  $\mu(73)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 293,\ (0:\ ),\ -0.681 - 0.732i)$
$L(\chi,\frac{1}{2})$  $\approx$  $-0.1428049747 + 0.3279397654i$
$L(\frac12,\chi)$  $\approx$  $-0.1428049747 + 0.3279397654i$
$L(\chi,1)$  $\approx$  0.3880942797 + 0.3961889561i
$L(1,\chi)$  $\approx$  0.3880942797 + 0.3961889561i

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

−24.904610743214936633954455435895, −24.38837087697542867384596927071, −23.00820385453510778413281964394, −22.158959789470308070094842365993, −20.973322293132155298758565308587, −19.94938429958732293028810984188, −19.57873998335132799009571416155, −18.56337811998134033765628317772, −17.95268008448274299226729478732, −16.785288106789768952365395759574, −16.02656243307796625708032107299, −14.7544736715279612396157187421, −13.20190179555843010479246743970, −12.691644312129249600099303073, −11.831463626178679910650193771037, −11.13567466859493638358198684892, −9.48434534357369183961049479846, −8.56935582178501382270066699031, −8.19379693132090159976059301859, −6.87498346148048167896917206579, −5.70513960420669133411111329970, −3.94170822878718637780524015614, −2.821532238463145717952796884636, −1.64659483138737980007749581258, −0.2875806470983334696724239296, 2.041013505512345788502680382175, 3.85950433836891695297125751630, 4.396365747525874554588669337704, 6.1722719909708881159452864270, 6.953723298011361730402035177737, 7.989083759560712209254948293, 9.140764752056973559154665795748, 9.91392666366706594012250589640, 10.889476585002116818569526802749, 11.457892005811213729070753867576, 13.480745015304258954755111254246, 14.5653101367547475101239710987, 15.01786623249081619627340579453, 16.022935340416714738265469705270, 16.80475191874777422250843829010, 17.561274489172712905898372991399, 18.85027299986241679491075984083, 19.80482408129778833540065341046, 20.10310885758982634574431011279, 21.64950485936335886526540058882, 22.593611969220972036670403146616, 23.354301662581617228715394435367, 24.16554201649702810275483525699, 25.7782697319456468123579420067, 26.01105773350285936993168790982

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