Properties

Degree 1
Conductor 283
Sign $1$
Motivic weight 0
Primitive yes
Self-dual yes
Analytic rank 0

Related objects

Learn more about

Normalization:  

Dirichlet series

L(χ,s)  = 1  − 2-s − 3-s + 4-s − 5-s + 6-s + 7-s − 8-s + 9-s + 10-s + 11-s − 12-s + 13-s − 14-s + 15-s + 16-s − 17-s − 18-s − 19-s − 20-s − 21-s − 22-s + 23-s + 24-s + 25-s − 26-s − 27-s + 28-s + ⋯
L(s,χ)  = 1  − 2-s − 3-s + 4-s − 5-s + 6-s + 7-s − 8-s + 9-s + 10-s + 11-s − 12-s + 13-s − 14-s + 15-s + 16-s − 17-s − 18-s − 19-s − 20-s − 21-s − 22-s + 23-s + 24-s + 25-s − 26-s − 27-s + 28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & \, \Lambda(\chi,1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & \, \Lambda(1-s,\chi) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(283\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{283} (282, \cdot )$
Sato-Tate  :  $\mu(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(1,\ 283,\ (1:\ ),\ 1)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.7962311769$
$L(\frac12,\chi)$  $\approx$  $0.7962311769$
$L(\chi,1)$  $\approx$  0.5602448972
$L(1,\chi)$  $\approx$  0.5602448972

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

−25.31242222675621664431551739755, −24.36610850506113872494423921514, −23.72258772846889737475407982539, −22.83723049365914314285573862869, −21.592323187791709648421065967413, −20.742393195445823091931121159737, −19.67382395362042506777090354142, −18.85039000295176749194293212797, −17.91300960854373126727963838780, −17.24394728289450202837672884657, −16.332277647475108549071836492664, −15.50514563989933891874055081686, −14.65436184856480236258973630043, −12.82744108125370481899881127449, −11.745963593102807174415108490889, −11.18778413704913020983330402150, −10.620249579466489680492312424333, −9.00335779439256028902621030802, −8.27094561702072828724535719141, −7.05196956742698477295442976043, −6.36740108023306933200730068547, −4.85415067204300359203549781994, −3.73900693182454968811093040255, −1.74695240500948440151273778144, −0.69556002559801935321157760296, 0.69556002559801935321157760296, 1.74695240500948440151273778144, 3.73900693182454968811093040255, 4.85415067204300359203549781994, 6.36740108023306933200730068547, 7.05196956742698477295442976043, 8.27094561702072828724535719141, 9.00335779439256028902621030802, 10.620249579466489680492312424333, 11.18778413704913020983330402150, 11.745963593102807174415108490889, 12.82744108125370481899881127449, 14.65436184856480236258973630043, 15.50514563989933891874055081686, 16.332277647475108549071836492664, 17.24394728289450202837672884657, 17.91300960854373126727963838780, 18.85039000295176749194293212797, 19.67382395362042506777090354142, 20.742393195445823091931121159737, 21.592323187791709648421065967413, 22.83723049365914314285573862869, 23.72258772846889737475407982539, 24.36610850506113872494423921514, 25.31242222675621664431551739755

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