Properties

Degree 1
Conductor $ 2^{5} \cdot 5^{3} $
Sign $0.244 + 0.969i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

Learn more about

Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.509 − 0.860i)3-s + (0.309 − 0.951i)7-s + (−0.481 − 0.876i)9-s + (0.562 + 0.827i)11-s + (−0.278 + 0.960i)13-s + (0.368 + 0.929i)17-s + (0.509 + 0.860i)19-s + (−0.661 − 0.750i)21-s + (−0.425 − 0.904i)23-s + (−0.999 − 0.0314i)27-s + (0.995 − 0.0941i)29-s + (−0.929 + 0.368i)31-s + (0.998 − 0.0627i)33-s + (−0.0314 − 0.999i)37-s + (0.684 + 0.728i)39-s + ⋯
L(s,χ)  = 1  + (0.509 − 0.860i)3-s + (0.309 − 0.951i)7-s + (−0.481 − 0.876i)9-s + (0.562 + 0.827i)11-s + (−0.278 + 0.960i)13-s + (0.368 + 0.929i)17-s + (0.509 + 0.860i)19-s + (−0.661 − 0.750i)21-s + (−0.425 − 0.904i)23-s + (−0.999 − 0.0314i)27-s + (0.995 − 0.0941i)29-s + (−0.929 + 0.368i)31-s + (0.998 − 0.0627i)33-s + (−0.0314 − 0.999i)37-s + (0.684 + 0.728i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.244 + 0.969i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.244 + 0.969i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4000\)    =    \(2^{5} \cdot 5^{3}\)
\( \varepsilon \)  =  $0.244 + 0.969i$
motivic weight  =  \(0\)
character  :  $\chi_{4000} (1117, \cdot )$
Sato-Tate  :  $\mu(200)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 4000,\ (1:\ ),\ 0.244 + 0.969i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.091556579 + 0.8508250127i$
$L(\frac12,\chi)$  $\approx$  $1.091556579 + 0.8508250127i$
$L(\chi,1)$  $\approx$  1.167229464 - 0.2659814410i
$L(1,\chi)$  $\approx$  1.167229464 - 0.2659814410i

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

−18.186837651051114394031210343243, −17.455655520365316989924264874607, −16.72984642734010241458723169451, −15.869303660927496739173792954421, −15.542224879475993262057203332601, −14.88228686618862187633887464638, −14.05773026524906082253386307956, −13.69001127378551539539160665414, −12.69635042665910087301095885135, −11.73139302271497009658254853808, −11.39815682544829102330168121671, −10.52297784193228933847307880591, −9.6638968794465696831126326686, −9.24320623233948742250543930862, −8.45381642199999206506973902723, −7.95006778583938381793583648183, −7.02059790157233401146051898293, −5.89351477921852250657392909512, −5.30326105995724818530703494246, −4.793404274772418407125292986376, −3.66350868586387138330031514244, −3.03838898114285076788869677277, −2.48025750101461348356088706952, −1.32467756868312172344407413353, −0.18578990832425161726521184610, 1.00897406787953263458070276376, 1.65409596215118004799482216251, 2.26848494464351910842107543414, 3.50545780980100199102483158500, 4.01715606932674030067927322062, 4.82748442453748694350817379872, 6.00638332927925451961822290562, 6.66134008345142473457001219385, 7.27848000074693363056614990947, 7.86942940412532061913769701031, 8.60383615624549821746291346504, 9.4042616658991571637831590914, 10.11205268613311501852197328720, 10.86317002063840015715185881804, 11.84557390552564940549678355619, 12.33571837256427954675185793578, 12.89255548017745454261494474579, 13.938410120482556267065391069057, 14.29102452694625352167908014563, 14.66259234875059766760576697745, 15.701876258759851697319471257238, 16.72015152983785883117961039408, 17.06639142228086967757256008457, 17.84168455314471163877487111075, 18.46681341765553499398611260620

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