↑

Properties

Label 1-4003-4003.3180-r0-0-0
Degree $1$
Conductor $4003$
Sign $0.329 + 0.944i$
Analytic cond. $18.5898$
Root an. cond. $18.5898$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(4003\)
Sign: $0.329 + 0.944i$
Analytic conductor: \(18.5898\)
Root analytic conductor: \(18.5898\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4003} (3180, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4003,\ (0:\ ),\ 0.329 + 0.944i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2127565215 + 0.1510720657i\)
\(L(\frac12)\) \(\approx\) \(0.2127565215 + 0.1510720657i\)
\(L(1)\) \(\approx\) \(0.5272074933 - 0.1825072293i\)
\(L(1)\) \(\approx\) \(0.5272074933 - 0.1825072293i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4003 \( 1 \)
good2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + T \)
13 \( 1 + T \)
17 \( 1 + T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 + (-0.5 + 0.866i)T \)
59 \( 1 + (-0.5 + 0.866i)T \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 + T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (-0.5 - 0.866i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (-0.5 - 0.866i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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