Properties

Degree 1
Conductor 73
Sign $0.239 - 0.970i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

Learn more about

Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.5 − 0.866i)2-s + 3-s + (−0.5 + 0.866i)4-s + (0.5 − 0.866i)5-s + (−0.5 − 0.866i)6-s − 7-s + 8-s + 9-s − 10-s + (0.5 − 0.866i)11-s + (−0.5 + 0.866i)12-s + (0.5 + 0.866i)13-s + (0.5 + 0.866i)14-s + (0.5 − 0.866i)15-s + (−0.5 − 0.866i)16-s − 17-s + ⋯
L(s,χ)  = 1  + (−0.5 − 0.866i)2-s + 3-s + (−0.5 + 0.866i)4-s + (0.5 − 0.866i)5-s + (−0.5 − 0.866i)6-s − 7-s + 8-s + 9-s − 10-s + (0.5 − 0.866i)11-s + (−0.5 + 0.866i)12-s + (0.5 + 0.866i)13-s + (0.5 + 0.866i)14-s + (0.5 − 0.866i)15-s + (−0.5 − 0.866i)16-s − 17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.239 - 0.970i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.239 - 0.970i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(73\)
\( \varepsilon \)  =  $0.239 - 0.970i$
motivic weight  =  \(0\)
character  :  $\chi_{73} (65, \cdot )$
Sato-Tate  :  $\mu(6)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 73,\ (0:\ ),\ 0.239 - 0.970i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.7656874079 - 0.5995306699i$
$L(\frac12,\chi)$  $\approx$  $0.7656874079 - 0.5995306699i$
$L(\chi,1)$  $\approx$  0.9245121981 - 0.4790706155i
$L(1,\chi)$  $\approx$  0.9245121981 - 0.4790706155i

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

−32.099032798787435620463272814637, −30.804173508733879047376524438253, −29.71235258495290283776103841420, −28.34614913482558434409375780884, −27.00461593996610892750672417990, −26.131549575870920058331012173924, −25.43314241340119034964920358950, −24.73301501007152174714058628247, −22.98810660433955330350634546869, −22.25397631419724303370326558068, −20.454409677789362549896016459281, −19.36035506379602856189949156666, −18.47665632468659773137332794486, −17.38386996068492324251831270325, −15.793399634671186168448255192481, −15.00898580413770221945996123057, −13.9603840664859350278739005229, −12.91960589919806125765655115866, −10.37134585204489098959629680352, −9.70322470319051802629175582411, −8.42388537586946058792868770137, −7.04795846411878612084115446536, −6.16813496892749252060058107898, −3.97943513142008565933036124605, −2.219104655344750449821747422474, 1.54923301212535917158709157097, 3.08196598809195255876574170965, 4.36873085138529348986222814178, 6.711749861270949985293823503650, 8.69973368001496644826859426901, 9.0038655152073136055451239223, 10.24835574580667601930760961169, 11.93140339447938320658403792317, 13.331645072784931625326162687309, 13.66550808950806051082731416374, 15.8428722907114946086464307781, 16.82487100279184954325719423129, 18.283603797005027896556688919796, 19.54956470268719183564745174164, 19.947417096859177684277441771146, 21.370759018255376997939121547353, 21.87958476914574804032240681764, 23.84995371498973916050326848444, 25.19566715032708781222068798971, 25.98758861295655644468801419129, 26.95384868415264421254446536425, 28.25537364259486958942917711487, 29.16568187053495023760639484818, 30.086679076921554972914625775610, 31.37594646283536642210910192232

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