Properties

Degree 1
Conductor 293
Sign $-0.579 + 0.815i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.0215 + 0.999i)2-s + (−0.234 − 0.972i)3-s + (−0.999 + 0.0430i)4-s + (−0.548 − 0.835i)5-s + (0.966 − 0.255i)6-s + (−0.991 − 0.128i)7-s + (−0.0645 − 0.997i)8-s + (−0.890 + 0.455i)9-s + (0.823 − 0.566i)10-s + (−0.618 + 0.785i)11-s + (0.276 + 0.961i)12-s + (0.869 − 0.493i)13-s + (0.107 − 0.994i)14-s + (−0.683 + 0.729i)15-s + (0.996 − 0.0859i)16-s + (0.0215 + 0.999i)17-s + ⋯
L(s,χ)  = 1  + (0.0215 + 0.999i)2-s + (−0.234 − 0.972i)3-s + (−0.999 + 0.0430i)4-s + (−0.548 − 0.835i)5-s + (0.966 − 0.255i)6-s + (−0.991 − 0.128i)7-s + (−0.0645 − 0.997i)8-s + (−0.890 + 0.455i)9-s + (0.823 − 0.566i)10-s + (−0.618 + 0.785i)11-s + (0.276 + 0.961i)12-s + (0.869 − 0.493i)13-s + (0.107 − 0.994i)14-s + (−0.683 + 0.729i)15-s + (0.996 − 0.0859i)16-s + (0.0215 + 0.999i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 293 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.579 + 0.815i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 293 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.579 + 0.815i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(293\)
\( \varepsilon \)  =  $-0.579 + 0.815i$
motivic weight  =  \(0\)
character  :  $\chi_{293} (69, \cdot )$
Sato-Tate  :  $\mu(73)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 293,\ (0:\ ),\ -0.579 + 0.815i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.1628870258 + 0.3156163390i$
$L(\frac12,\chi)$  $\approx$  $0.1628870258 + 0.3156163390i$
$L(\chi,1)$  $\approx$  0.5786886000 + 0.1290216756i
$L(1,\chi)$  $\approx$  0.5786886000 + 0.1290216756i

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

−25.72485754909553786184415540337, −23.609898662109048748464872160368, −23.12284698392443921391036749671, −22.20865425136495844224437031135, −21.66756289633014123802576812366, −20.69962893208956716007604870368, −19.78058316226070746575857066724, −18.82658393807010103471965347305, −18.278506604900300330033652097977, −16.809360716444496089825970649478, −15.95084397241605666544410509229, −15.0578954902106971643045652086, −13.94398892745965958268563434035, −13.02060420831094394325235938399, −11.67697138315144318683972850884, −11.043809096393636326687507863858, −10.35812730025772879085453442734, −9.28369030396534354692861754452, −8.500636853649427104242823487001, −6.76701371661692964092378490087, −5.56802350763477386546081150778, −4.32087678480234011747597492616, −3.30020918559277261907384913282, −2.75476188698134342189756385274, −0.26489464052788452902153473158, 1.288528433076681182723649411186, 3.33132311764424981605976818889, 4.61499167763418555367189742585, 5.808509277031809180899882713184, 6.5539190200221119195198080513, 7.74050203563739375494965643137, 8.28584536100095172554872273638, 9.43085817972386562135781836392, 10.759956759481952196128667677881, 12.449721614043776131213308635477, 12.81424246328338863137705356770, 13.49434261083027203380904298931, 14.92946901408442852423539997444, 15.786414088466854117901765039260, 16.68645619630046988645185548463, 17.36713483656994155015000995747, 18.42799793487403368065209767196, 19.22828341370804312088291660074, 20.063929393955652946562552708815, 21.38400711479157758293889999030, 22.9776663098567037772657190006, 23.1242652241814225117354421300, 23.83408255354423178376096851532, 25.06052826745906607381689435653, 25.38231410586294314372951579708

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