Properties

Degree 1
Conductor 293
Sign $0.822 - 0.568i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.0645 − 0.997i)2-s + (0.651 + 0.758i)3-s + (−0.991 + 0.128i)4-s + (0.985 − 0.171i)5-s + (0.714 − 0.699i)6-s + (−0.925 − 0.377i)7-s + (0.192 + 0.981i)8-s + (−0.150 + 0.988i)9-s + (−0.234 − 0.972i)10-s + (0.908 + 0.417i)11-s + (−0.744 − 0.668i)12-s + (0.0215 − 0.999i)13-s + (−0.317 + 0.948i)14-s + (0.772 + 0.635i)15-s + (0.966 − 0.255i)16-s + (−0.0645 − 0.997i)17-s + ⋯
L(s,χ)  = 1  + (−0.0645 − 0.997i)2-s + (0.651 + 0.758i)3-s + (−0.991 + 0.128i)4-s + (0.985 − 0.171i)5-s + (0.714 − 0.699i)6-s + (−0.925 − 0.377i)7-s + (0.192 + 0.981i)8-s + (−0.150 + 0.988i)9-s + (−0.234 − 0.972i)10-s + (0.908 + 0.417i)11-s + (−0.744 − 0.668i)12-s + (0.0215 − 0.999i)13-s + (−0.317 + 0.948i)14-s + (0.772 + 0.635i)15-s + (0.966 − 0.255i)16-s + (−0.0645 − 0.997i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 293 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.822 - 0.568i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 293 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.822 - 0.568i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(293\)
\( \varepsilon \)  =  $0.822 - 0.568i$
motivic weight  =  \(0\)
character  :  $\chi_{293} (56, \cdot )$
Sato-Tate  :  $\mu(73)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 293,\ (0:\ ),\ 0.822 - 0.568i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.462543834 - 0.4565442683i$
$L(\frac12,\chi)$  $\approx$  $1.462543834 - 0.4565442683i$
$L(\chi,1)$  $\approx$  1.242882327 - 0.3282497030i
$L(1,\chi)$  $\approx$  1.242882327 - 0.3282497030i

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

−25.53957093477825962240936508827, −24.62721275417563541402933348950, −24.238498846944385632824942559322, −22.940046366977530718534881392030, −22.09873010124134269203797133089, −21.2761008248654817125238837471, −19.74702723597077248519157395880, −18.946231692091939502854161196008, −18.35129738061621662216674625920, −17.2004835475722060695933655017, −16.60579631832594883516490879593, −15.21870159038244998916295407087, −14.48401628404988259099665518809, −13.53352651585846412215012070538, −13.10405263744639442991127437412, −11.84294079602462144560619637875, −9.99495667199659413799630106865, −9.14129070272222531358702909660, −8.63798930923464574469698057402, −7.071620642212105055094808446438, −6.48529633334172830410965175069, −5.75824749235439686751126534134, −4.037599043733927102946421560910, −2.76564605816394629155596970439, −1.28674888825918193319788388245, 1.32558336947291899011242861342, 2.75837764904980635023724075051, 3.46298538850015618972878098078, 4.70447098427267356564908049041, 5.73809079438873831136173702875, 7.42811528093948598739469492081, 8.85773490881664523192418534767, 9.65075727858917385272762809956, 9.98269774541275559123270475095, 11.11452056628403011551424757518, 12.45774332822023299385956095014, 13.4167906212911317075755447327, 13.97404086236575478899778744957, 15.04578997068026401527597492753, 16.403417358380868739221639819310, 17.18459249111026873065394882618, 18.23389161690753906082201749017, 19.33494139699733400355776523897, 20.27895698431545303940835039775, 20.54671414123982020063362780559, 21.735710852790273432357611662622, 22.397062203076803080572611293485, 22.99918308211281098341506903266, 24.97279406509340464872007795322, 25.36799179073870586514815180208

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