Properties

Degree 1
Conductor 293
Sign $0.834 + 0.551i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

Learn more about

Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.683 + 0.729i)2-s + (0.908 + 0.417i)3-s + (−0.0645 − 0.997i)4-s + (0.996 − 0.0859i)5-s + (−0.925 + 0.377i)6-s + (0.192 − 0.981i)7-s + (0.772 + 0.635i)8-s + (0.651 + 0.758i)9-s + (−0.618 + 0.785i)10-s + (−0.976 − 0.213i)11-s + (0.357 − 0.933i)12-s + (0.714 − 0.699i)13-s + (0.584 + 0.811i)14-s + (0.941 + 0.337i)15-s + (−0.991 + 0.128i)16-s + (−0.683 + 0.729i)17-s + ⋯
L(s,χ)  = 1  + (−0.683 + 0.729i)2-s + (0.908 + 0.417i)3-s + (−0.0645 − 0.997i)4-s + (0.996 − 0.0859i)5-s + (−0.925 + 0.377i)6-s + (0.192 − 0.981i)7-s + (0.772 + 0.635i)8-s + (0.651 + 0.758i)9-s + (−0.618 + 0.785i)10-s + (−0.976 − 0.213i)11-s + (0.357 − 0.933i)12-s + (0.714 − 0.699i)13-s + (0.584 + 0.811i)14-s + (0.941 + 0.337i)15-s + (−0.991 + 0.128i)16-s + (−0.683 + 0.729i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 293 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.834 + 0.551i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 293 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.834 + 0.551i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(293\)
\( \varepsilon \)  =  $0.834 + 0.551i$
motivic weight  =  \(0\)
character  :  $\chi_{293} (39, \cdot )$
Sato-Tate  :  $\mu(73)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 293,\ (0:\ ),\ 0.834 + 0.551i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.369367785 + 0.4118140988i$
$L(\frac12,\chi)$  $\approx$  $1.369367785 + 0.4118140988i$
$L(\chi,1)$  $\approx$  1.147939104 + 0.3175228547i
$L(1,\chi)$  $\approx$  1.147939104 + 0.3175228547i

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

−25.51005619102719500635575816146, −24.888464105841301632429550078471, −23.81561236141757538913027512648, −22.173865801772340575302895158391, −21.5224004093629754733810548608, −20.70328073448151406601443585632, −20.09013224598988077791479838822, −18.739164346213674571596260431176, −18.28497072445431638606639320477, −17.77944589689403254214490373888, −16.22646926122619604089502876800, −15.3372365702151051376785466249, −13.85739302628521843299187661736, −13.40452809856164371306923774265, −12.343372852768245804958719414114, −11.35427354137262790921135767089, −10.03941774798200184387182440266, −9.2737348528192624495738788050, −8.59664252743442426779724416887, −7.52495756175847118687566138274, −6.356991247487279816227524787967, −4.825006376250191488655585458942, −3.15047279507926290822597742104, −2.36000474009144777560104156741, −1.53376142124470000204839249858, 1.29411303713614248694086835209, 2.56302209143622603709255915589, 4.168922040396671806761481667347, 5.35656171322716127103361009380, 6.415990782488944733001128401396, 7.802396817588902490609833662492, 8.27915915207957842399454853530, 9.5239810827375931178897112844, 10.263611345149657594831619762130, 10.83723067833732272209133926589, 13.204589556900519297716049392048, 13.659186632895251483605994482460, 14.50260897847114156605562911483, 15.6006559687377374942164935787, 16.2827443621734726551420559029, 17.38637942890831005220987309581, 18.11123483449034371545378934434, 19.07784121744420705693071009385, 20.39979141023543586037381507448, 20.533738446076568122486837715280, 21.8544871366292782059565808703, 23.01573190921945178509359826357, 24.13459868931556026348573487164, 24.80751922633299978467328054906, 25.73649600847024506455015642927

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