Properties

Degree 1
Conductor 23
Sign $0.381 - 0.924i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.959 + 0.281i)2-s + (−0.654 − 0.755i)3-s + (0.841 − 0.540i)4-s + (−0.142 − 0.989i)5-s + (0.841 + 0.540i)6-s + (0.415 − 0.909i)7-s + (−0.654 + 0.755i)8-s + (−0.142 + 0.989i)9-s + (0.415 + 0.909i)10-s + (−0.959 − 0.281i)11-s + (−0.959 − 0.281i)12-s + (0.415 + 0.909i)13-s + (−0.142 + 0.989i)14-s + (−0.654 + 0.755i)15-s + (0.415 − 0.909i)16-s + (0.841 + 0.540i)17-s + ⋯
L(s,χ)  = 1  + (−0.959 + 0.281i)2-s + (−0.654 − 0.755i)3-s + (0.841 − 0.540i)4-s + (−0.142 − 0.989i)5-s + (0.841 + 0.540i)6-s + (0.415 − 0.909i)7-s + (−0.654 + 0.755i)8-s + (−0.142 + 0.989i)9-s + (0.415 + 0.909i)10-s + (−0.959 − 0.281i)11-s + (−0.959 − 0.281i)12-s + (0.415 + 0.909i)13-s + (−0.142 + 0.989i)14-s + (−0.654 + 0.755i)15-s + (0.415 − 0.909i)16-s + (0.841 + 0.540i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.381 - 0.924i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 23 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.381 - 0.924i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(23\)
\( \varepsilon \)  =  $0.381 - 0.924i$
motivic weight  =  \(0\)
character  :  $\chi_{23} (3, \cdot )$
Sato-Tate  :  $\mu(11)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 23,\ (0:\ ),\ 0.381 - 0.924i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.3277273320 - 0.2192356379i$
$L(\frac12,\chi)$  $\approx$  $0.3277273320 - 0.2192356379i$
$L(\chi,1)$  $\approx$  0.5201889752 - 0.1855913550i
$L(1,\chi)$  $\approx$  0.5201889752 - 0.1855913550i

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

−38.86901206720230220250172177424, −38.04978520853081323510024006482, −37.07764318936724965380662137499, −35.09279704228361515166147371397, −34.34335952807736822061983349341, −33.43445248621830591262689882652, −31.32850806200926789795211473721, −29.85958297183451124905676931097, −28.54104978554951604586887868863, −27.526893336742595010903382585128, −26.497839210692829050247391031, −25.19677779353309866265573337361, −23.07320963138608532764277937631, −21.69860673246791804470920090761, −20.58110386643468482917126950490, −18.52349463683331496984900599528, −17.87536071873842841425803508647, −16.02772464174142797019205408389, −15.06133768301543924413000424979, −12.0787049912967822951052893076, −10.88483862655997164205213863176, −9.768718420132840337553343315027, −7.81320610194552651361795590287, −5.78010856069087742744357667992, −3.01197442570287685806567374878, 1.27653878245145980082795619280, 5.31655872189447821089738407592, 7.19217904241057344185175816264, 8.43825346336987519118210847845, 10.52002687418432590322427583875, 11.92528411857715341626447780780, 13.69354664177652326072264669422, 16.125151922336169579796691386233, 16.99597997769593897315678826565, 18.24509377294006263087636494940, 19.62895355377679179633632731038, 20.98126750068120979775972839618, 23.746162420970419735183934618656, 23.90697140200119864816739964174, 25.58753454839631486009922192740, 27.14720500155539862608126843923, 28.45233492006461977443780344521, 29.195470169184944176971213074104, 30.723716614579539498499726230653, 32.78621332865303291066825772579, 33.97287555764827752484027603333, 35.13830079197997026787697269749, 36.345291181608239217906100369, 36.76342345208191279745533848015, 38.98998143260861314826998406865

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