Properties

Degree 1
Conductor 293
Sign $-0.183 + 0.983i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

Learn more about

Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.584 + 0.811i)2-s + (−0.548 − 0.835i)3-s + (−0.317 + 0.948i)4-s + (0.908 + 0.417i)5-s + (0.357 − 0.933i)6-s + (0.823 + 0.566i)7-s + (−0.954 + 0.296i)8-s + (−0.397 + 0.917i)9-s + (0.192 + 0.981i)10-s + (−0.474 + 0.880i)11-s + (0.966 − 0.255i)12-s + (−0.744 − 0.668i)13-s + (0.0215 + 0.999i)14-s + (−0.150 − 0.988i)15-s + (−0.798 − 0.601i)16-s + (0.584 + 0.811i)17-s + ⋯
L(s,χ)  = 1  + (0.584 + 0.811i)2-s + (−0.548 − 0.835i)3-s + (−0.317 + 0.948i)4-s + (0.908 + 0.417i)5-s + (0.357 − 0.933i)6-s + (0.823 + 0.566i)7-s + (−0.954 + 0.296i)8-s + (−0.397 + 0.917i)9-s + (0.192 + 0.981i)10-s + (−0.474 + 0.880i)11-s + (0.966 − 0.255i)12-s + (−0.744 − 0.668i)13-s + (0.0215 + 0.999i)14-s + (−0.150 − 0.988i)15-s + (−0.798 − 0.601i)16-s + (0.584 + 0.811i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 293 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.183 + 0.983i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 293 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.183 + 0.983i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(293\)
\( \varepsilon \)  =  $-0.183 + 0.983i$
motivic weight  =  \(0\)
character  :  $\chi_{293} (81, \cdot )$
Sato-Tate  :  $\mu(73)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 293,\ (0:\ ),\ -0.183 + 0.983i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.9623886972 + 1.158164686i$
$L(\frac12,\chi)$  $\approx$  $0.9623886972 + 1.158164686i$
$L(\chi,1)$  $\approx$  1.137821770 + 0.6353803898i
$L(1,\chi)$  $\approx$  1.137821770 + 0.6353803898i

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

−24.88490164166294636004550399137, −24.10216513971510690182810010477, −23.220641362226413144308364211724, −22.31528846375399607756597515280, −21.31130263503534040733585160297, −21.03018253863651928887814468447, −20.29711788297499034921212323090, −18.86411706107123598403397711441, −17.94058627682383383060908200859, −16.89965258652007456812567833862, −16.20749218209895030405567181962, −14.615307585420440073130937232798, −14.23767316910599455216680470185, −13.119064256289325462679998402336, −11.97780351321775469659239937405, −11.17546105044860996740782831401, −10.234726198600206931317461421309, −9.62550253103121499236961215905, −8.43533082573102165188007410917, −6.51860223027745743672877070531, −5.36546170642645940876919414743, −4.87832164437162508273921637843, −3.74528513590669221131530325886, −2.351152374671433959956664146874, −0.91512879777838158265598933079, 1.85404165669837155814648286516, 2.85784770302617202207106951443, 4.93638478714520681009068920613, 5.40194252253615351646278913511, 6.448649125217087082475272759621, 7.41428200141131014877697551478, 8.17985458620567036237892231178, 9.61206300251412793774979971728, 10.954105013629278854772122877263, 12.051120989560519614159318287, 12.89974503942898089269640052504, 13.60306858480140363704160724009, 14.78251665822461143284373268332, 15.258076977782868548552157021544, 16.89270508222153403081885807618, 17.56555105099420646437312525820, 17.95094454211844680606353269665, 19.01488589665218431606990794882, 20.56605541350137093768069559194, 21.6246148758816340637889277120, 22.18822020563683520544495972651, 23.141149608637390602487923024080, 23.99520161024118213340267898673, 24.72739404062567056363865586464, 25.47014857877601887787383514754

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