Properties

Degree 1
Conductor 293
Sign $0.840 - 0.541i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.890 + 0.455i)2-s + (−0.474 − 0.880i)3-s + (0.584 − 0.811i)4-s + (−0.976 + 0.213i)5-s + (0.823 + 0.566i)6-s + (−0.954 + 0.296i)7-s + (−0.150 + 0.988i)8-s + (−0.548 + 0.835i)9-s + (0.772 − 0.635i)10-s + (0.512 − 0.858i)11-s + (−0.991 − 0.128i)12-s + (0.357 + 0.933i)13-s + (0.714 − 0.699i)14-s + (0.651 + 0.758i)15-s + (−0.317 − 0.948i)16-s + (−0.890 + 0.455i)17-s + ⋯
L(s,χ)  = 1  + (−0.890 + 0.455i)2-s + (−0.474 − 0.880i)3-s + (0.584 − 0.811i)4-s + (−0.976 + 0.213i)5-s + (0.823 + 0.566i)6-s + (−0.954 + 0.296i)7-s + (−0.150 + 0.988i)8-s + (−0.548 + 0.835i)9-s + (0.772 − 0.635i)10-s + (0.512 − 0.858i)11-s + (−0.991 − 0.128i)12-s + (0.357 + 0.933i)13-s + (0.714 − 0.699i)14-s + (0.651 + 0.758i)15-s + (−0.317 − 0.948i)16-s + (−0.890 + 0.455i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 293 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.840 - 0.541i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 293 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.840 - 0.541i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(293\)
\( \varepsilon \)  =  $0.840 - 0.541i$
motivic weight  =  \(0\)
character  :  $\chi_{293} (65, \cdot )$
Sato-Tate  :  $\mu(73)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 293,\ (0:\ ),\ 0.840 - 0.541i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.4161037821 - 0.1224224841i$
$L(\frac12,\chi)$  $\approx$  $0.4161037821 - 0.1224224841i$
$L(\chi,1)$  $\approx$  0.4789530499 - 0.03421916366i
$L(1,\chi)$  $\approx$  0.4789530499 - 0.03421916366i

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

−25.86373433433792113547768158696, −24.93209468703762174943512646363, −23.504987754450713668552308844, −22.60373513324275527293582346306, −22.062525754644186955124892061099, −20.65562850110070227813543342387, −20.0350156772201101219863337601, −19.53332461307084042955862357199, −18.09891788910611623659977455130, −17.379174045538511105201800933226, −16.291079469439964491513429841, −15.810100081954700190577408963247, −15.01523084061649052281617303480, −13.06646792273059236741923910501, −12.24938694486940324411760055279, −11.261243643260388211350990755805, −10.59567492550996152496426319231, −9.43733905566911835300833475179, −8.92063351569551794641506654169, −7.43225632511960325505503267603, −6.644085513905767229787269449442, −4.94478791400300582340670098792, −3.75156118548994784629436594349, −3.03339796244865765997839430518, −0.8287508942255349209955633608, 0.61689989191219786225109842528, 2.15809704165263629126133348144, 3.66633749979129581665222856614, 5.50210037571539346961740905103, 6.597026560800792704383698357360, 6.98113114445829159248750856555, 8.31720221453820234231054114433, 8.93841231392137005612729064024, 10.431633310337769567969423335471, 11.42697588965372073250460696215, 11.991781005455610112744902292878, 13.33205463634607495970066363070, 14.42252444672198248970987919785, 15.62973841938037515403004540342, 16.396902915325331872959420940545, 17.026640021368696638295595087416, 18.35045014781143200466084866268, 19.0733600710316422740806277330, 19.29281307760576626390507491253, 20.42953724582858652780387346360, 22.14442286110158546878835633768, 22.90140453933282887904936193889, 23.84308941857982404969896124257, 24.423020702579850855177445117194, 25.334626520796428986547362518262

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