Properties

Degree 1
Conductor 4003
Sign $0.329 - 0.944i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

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

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 4003 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.329 - 0.944i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 4003 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.329 - 0.944i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4003\)
\( \varepsilon \)  =  $0.329 - 0.944i$
motivic weight  =  \(0\)
character  :  $\chi_{4003} (822, \cdot )$
Sato-Tate  :  $\mu(3)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 4003,\ (0:\ ),\ 0.329 - 0.944i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.2127565215 - 0.1510720657i$
$L(\frac12,\chi)$  $\approx$  $0.2127565215 - 0.1510720657i$
$L(\chi,1)$  $\approx$  0.5272074933 + 0.1825072293i
$L(1,\chi)$  $\approx$  0.5272074933 + 0.1825072293i

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

−18.68987359763322336737826474944, −17.87480745974621749688222440803, −16.95481312103000432690442039443, −16.77831029810397178750194153492, −16.2486382795255493442293532173, −15.48984787255043459631441745250, −14.44694823406804337836622797794, −13.73337831591870032122750319709, −12.759590044613739735560035855097, −12.44986526151031680247025396016, −11.46023516343005651471573453449, −11.11942981169817554600231875691, −10.3860447641413391366159969386, −9.57604237540423216493852422634, −9.1509516526637716145602521700, −8.455308768193512091248491630826, −7.65274405358870295376441985411, −6.663343135617911488926428631070, −5.85334238107030221935445732939, −4.669047294982520743807465414769, −4.28172179863661821048005363922, −3.619071763013146054170725871807, −3.047576585454215775058836511418, −1.437814656058299221070415134309, −0.90719287920367153589592424808, 0.12335347063239846213269847171, 1.42210704521603171117109706083, 2.031177088540210360183405561011, 3.36378970146518672676866744857, 3.98287300228979392958342257472, 5.38306830642828173703325146893, 5.924679117852098219829810148758, 6.38631966691103189701277264505, 7.062662844288171621977918243666, 7.78288254426460125414827619249, 8.38633396190509332382468413706, 9.14314428359981251349443171205, 10.027134431137780097215347933879, 10.737867053111850485078512726711, 11.54689014725359654935436639546, 12.065273124118432070551568634, 12.889336138751505039065532397510, 13.78773718288865931662913878375, 14.365040750840318063558505770008, 14.94704534859128432804096217798, 15.82337209820049490477730023740, 16.29195707943596342677374423597, 17.00702627544740811956689920196, 17.818532541507736081491904765920, 18.33690841031133853907963442395

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