Properties

Degree 1
Conductor 73
Sign $0.998 - 0.0614i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

Learn more about

Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.939 + 0.342i)2-s + (0.5 − 0.866i)3-s + (0.766 − 0.642i)4-s + (−0.342 + 0.939i)5-s + (−0.173 + 0.984i)6-s + (0.866 + 0.5i)7-s + (−0.5 + 0.866i)8-s + (−0.5 − 0.866i)9-s i·10-s + (0.342 − 0.939i)11-s + (−0.173 − 0.984i)12-s + (0.984 + 0.173i)13-s + (−0.984 − 0.173i)14-s + (0.642 + 0.766i)15-s + (0.173 − 0.984i)16-s + (−0.866 + 0.5i)17-s + ⋯
L(s,χ)  = 1  + (−0.939 + 0.342i)2-s + (0.5 − 0.866i)3-s + (0.766 − 0.642i)4-s + (−0.342 + 0.939i)5-s + (−0.173 + 0.984i)6-s + (0.866 + 0.5i)7-s + (−0.5 + 0.866i)8-s + (−0.5 − 0.866i)9-s i·10-s + (0.342 − 0.939i)11-s + (−0.173 − 0.984i)12-s + (0.984 + 0.173i)13-s + (−0.984 − 0.173i)14-s + (0.642 + 0.766i)15-s + (0.173 − 0.984i)16-s + (−0.866 + 0.5i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.998 - 0.0614i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.998 - 0.0614i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(73\)
\( \varepsilon \)  =  $0.998 - 0.0614i$
motivic weight  =  \(0\)
character  :  $\chi_{73} (12, \cdot )$
Sato-Tate  :  $\mu(36)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 73,\ (0:\ ),\ 0.998 - 0.0614i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.7663966186 + 0.02355251800i$
$L(\frac12,\chi)$  $\approx$  $0.7663966186 + 0.02355251800i$
$L(\chi,1)$  $\approx$  0.8355444745 + 0.01032304419i
$L(1,\chi)$  $\approx$  0.8355444745 + 0.01032304419i

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

−31.177473798505921125510807397501, −30.75703357179634360332786780837, −29.063033753322010827602206475647, −27.89628345894253293498949098781, −27.48132323270061688794898535029, −26.414240729862731355113013603859, −25.27532223408318963126136446540, −24.3199796451943100816197070140, −22.61541764558229745508294599691, −20.9860289824625071745736452199, −20.49542268435335141438948182381, −19.808630535464056796246405237452, −18.13910910971505697179466189708, −16.97920682036604592189945738975, −16.06566371494271130774966229786, −14.988786846512179996140862762929, −13.31128519317833549247447258624, −11.7135921808415444899847549776, −10.70777799134582561155071434624, −9.35143703535484790655853692279, −8.532538162752212672756832304185, −7.3851792137668566839023312199, −4.9055965231168360149940450029, −3.64857587976075883950468799733, −1.62645926424466074053834059009, 1.592299349568072614860886335497, 3.12183658162530161909760570387, 5.9382791408685592145814931549, 7.01932316027153988905098817818, 8.18091306147684789337970562313, 9.025720977661862474603132256170, 11.018924522842500418409529486689, 11.623052497375575861157215652586, 13.70458243269948052348345136398, 14.73465933103154708464530841135, 15.703266507374664503042721716508, 17.45883612446192906736239947576, 18.35107679556160553526771080221, 19.00697220531294836680012893204, 20.06200661970700365591751660876, 21.44824083574992677831899538306, 23.23352407828415065965842334129, 24.25654816166284712562523859428, 25.03023311781580645168626086063, 26.257703871686222846676596353873, 26.90552644354690203727963948102, 28.22703456165460358518663597828, 29.40570579219387568399429757718, 30.44781734412241903372406922114, 31.17060189533640387618595094746

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