Properties

Degree 1
Conductor $ 2^{5} \cdot 5^{3} $
Sign $0.708 - 0.705i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

Learn more about

Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.940 + 0.338i)3-s + (−0.809 − 0.587i)7-s + (0.770 − 0.637i)9-s + (0.750 − 0.661i)11-s + (−0.0941 + 0.995i)13-s + (−0.125 − 0.992i)17-s + (−0.940 − 0.338i)19-s + (0.960 + 0.278i)21-s + (−0.929 − 0.368i)23-s + (−0.509 + 0.860i)27-s + (−0.999 + 0.0314i)29-s + (−0.992 + 0.125i)31-s + (−0.481 + 0.876i)33-s + (−0.860 + 0.509i)37-s + (−0.248 − 0.968i)39-s + ⋯
L(s,χ)  = 1  + (−0.940 + 0.338i)3-s + (−0.809 − 0.587i)7-s + (0.770 − 0.637i)9-s + (0.750 − 0.661i)11-s + (−0.0941 + 0.995i)13-s + (−0.125 − 0.992i)17-s + (−0.940 − 0.338i)19-s + (0.960 + 0.278i)21-s + (−0.929 − 0.368i)23-s + (−0.509 + 0.860i)27-s + (−0.999 + 0.0314i)29-s + (−0.992 + 0.125i)31-s + (−0.481 + 0.876i)33-s + (−0.860 + 0.509i)37-s + (−0.248 − 0.968i)39-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.708 - 0.705i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.708 - 0.705i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4000\)    =    \(2^{5} \cdot 5^{3}\)
\( \varepsilon \)  =  $0.708 - 0.705i$
motivic weight  =  \(0\)
character  :  $\chi_{4000} (373, \cdot )$
Sato-Tate  :  $\mu(200)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 4000,\ (1:\ ),\ 0.708 - 0.705i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.5052730049 - 0.2088261549i$
$L(\frac12,\chi)$  $\approx$  $0.5052730049 - 0.2088261549i$
$L(\chi,1)$  $\approx$  0.6121714050 + 0.0006755481933i
$L(1,\chi)$  $\approx$  0.6121714050 + 0.0006755481933i

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

−18.28319088911271095256499359666, −17.68333269730555356808204491917, −17.08775245136801555462298676707, −16.50807858878473684124983642136, −15.68324203634985201472004025276, −15.14246307349554648176138593118, −14.43158231864212728971825677417, −13.26778318673225317926479363473, −12.77646535771226547751111567793, −12.32360475004722532254425227482, −11.69232133594612017663908336334, −10.71563048564378202453765651453, −10.26933892744597218672766656188, −9.481495468560216427629594962075, −8.69275903816361671191040782638, −7.75863928318854418422937668507, −7.08965120899573292537832055830, −6.21941945424839275123952811804, −5.87530246701192444295001760335, −5.10958264842572959058598923914, −4.04064418594630000411572600305, −3.5086780293869873210183579218, −2.09976857545749369718067228531, −1.73775548232670621712201541926, −0.37695076595337323407753265863, 0.22500103995163801821433626007, 1.13391306832201000279378471182, 2.1523713327557143564582606450, 3.37733777100304053398153498248, 4.00590207608942919584812331890, 4.59586565861380813258117210451, 5.57464455676326256600243930629, 6.33578915316298751361042071294, 6.79494813185956194987656499483, 7.43894119695445187382477292798, 8.75039143984393023412548347482, 9.30273309619780411343833402558, 9.94727365324169792583077150951, 10.73528410554222240413657138274, 11.32172955048144800321532793176, 11.954293364133779284066636739545, 12.63782511913608804245690147754, 13.46360193966996402315187681560, 14.06146253350990461684688984173, 14.88003231002450082776397526917, 15.791953097068207183077326081031, 16.30735502850234586188851387966, 16.91684414711919143361432103615, 17.169051539575034933747765111402, 18.43888913731515819053417361948

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