↑

Properties

Label 1-4024-4024.1261-r0-0-0
Degree $1$
Conductor $4024$
Sign $0.740 + 0.671i$
Analytic cond. $18.6873$
Root an. cond. $18.6873$
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  + (−0.955 + 0.295i)3-s + (−0.118 + 0.992i)5-s + (0.695 + 0.718i)7-s + (0.824 − 0.565i)9-s + (0.277 − 0.960i)11-s + (0.570 − 0.821i)13-s + (−0.180 − 0.983i)15-s + (0.864 + 0.501i)17-s + (−0.0937 − 0.995i)19-s + (−0.877 − 0.479i)21-s + (−0.507 + 0.861i)23-s + (−0.971 − 0.235i)25-s + (−0.620 + 0.784i)27-s + (−0.998 − 0.0500i)29-s + (0.517 − 0.855i)31-s + ⋯
L(s)  = 1  + (−0.955 + 0.295i)3-s + (−0.118 + 0.992i)5-s + (0.695 + 0.718i)7-s + (0.824 − 0.565i)9-s + (0.277 − 0.960i)11-s + (0.570 − 0.821i)13-s + (−0.180 − 0.983i)15-s + (0.864 + 0.501i)17-s + (−0.0937 − 0.995i)19-s + (−0.877 − 0.479i)21-s + (−0.507 + 0.861i)23-s + (−0.971 − 0.235i)25-s + (−0.620 + 0.784i)27-s + (−0.998 − 0.0500i)29-s + (0.517 − 0.855i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.740 + 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.740 + 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4024\)    =    \(2^{3} \cdot 503\)
Sign: $0.740 + 0.671i$
Analytic conductor: \(18.6873\)
Root analytic conductor: \(18.6873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4024} (1261, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4024,\ (0:\ ),\ 0.740 + 0.671i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.283971481 + 0.4954014003i\)
\(L(\frac12)\) \(\approx\) \(1.283971481 + 0.4954014003i\)
\(L(1)\) \(\approx\) \(0.8898756588 + 0.2084669653i\)
\(L(1)\) \(\approx\) \(0.8898756588 + 0.2084669653i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
503 \( 1 \)
good3 \( 1 + (-0.955 + 0.295i)T \)
5 \( 1 + (-0.118 + 0.992i)T \)
7 \( 1 + (0.695 + 0.718i)T \)
11 \( 1 + (0.277 - 0.960i)T \)
13 \( 1 + (0.570 - 0.821i)T \)
17 \( 1 + (0.864 + 0.501i)T \)
19 \( 1 + (-0.0937 - 0.995i)T \)
23 \( 1 + (-0.507 + 0.861i)T \)
29 \( 1 + (-0.998 - 0.0500i)T \)
31 \( 1 + (0.517 - 0.855i)T \)
37 \( 1 + (0.131 - 0.991i)T \)
41 \( 1 + (-0.325 - 0.945i)T \)
43 \( 1 + (0.00625 + 0.999i)T \)
47 \( 1 + (0.560 + 0.828i)T \)
53 \( 1 + (-0.0937 + 0.995i)T \)
59 \( 1 + (-0.677 + 0.735i)T \)
61 \( 1 + (-0.0187 + 0.999i)T \)
67 \( 1 + (0.180 - 0.983i)T \)
71 \( 1 + (0.600 + 0.799i)T \)
73 \( 1 + (0.539 - 0.842i)T \)
79 \( 1 + (-0.106 - 0.994i)T \)
83 \( 1 + (0.999 - 0.0375i)T \)
89 \( 1 + (-0.925 - 0.378i)T \)
97 \( 1 + (-0.965 + 0.259i)T \)
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

−18.40593434323728529341251513897, −17.568875265466135922765174745409, −16.86154999382589649741302924155, −16.658766634993585151981687751987, −15.9596930575173101976267381311, −15.03507438901358272825734908391, −14.13193805538835937876181495845, −13.62197050425587822436310487517, −12.69472802744832050703043220855, −12.17957125757083499245849822819, −11.68036569531871223709834253017, −10.94991206520815937801937009669, −10.04130186436941333005835021496, −9.63666902626066761101927079521, −8.4171667445194118061274403962, −7.96355953872314878735047706272, −7.0844242244353856961719564835, −6.51612878547199570241319751046, −5.509300163409913778648736697779, −4.92151071082519625544790347799, −4.29377897490229973530754785983, −3.685288242184451231094573380653, −1.909301330082446110808247542990, −1.544071335402042040449184220556, −0.69221290403506376168777465946, 0.704857417957496718063912947903, 1.70325060367181287981640929293, 2.78255881108551146222270949304, 3.55554750635027442952041204625, 4.248569033494315344934369794783, 5.38066470347143218341731866279, 5.880539505676343073244540304268, 6.25688542269446854672647826570, 7.46147393801755440372166335577, 7.87302771527358491638900826090, 8.94348204922667323697355663839, 9.61862045048174206285526761813, 10.65766850296294895721030953179, 10.91394998113673076753386577225, 11.570831309645178120213144067295, 12.115456103780426780368859047674, 13.04315804974976115714136450140, 13.83422432627921785487929917172, 14.61965941406603247406665230699, 15.39669375724970614040099767694, 15.58457777925669502672048965873, 16.5707984429294346945704625573, 17.331986909348112262191185593360, 17.85554209145815621947706837052, 18.47343560683272346262409968376

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