↑

Properties

Label 1-4024-4024.61-r0-0-0
Degree $1$
Conductor $4024$
Sign $0.963 - 0.265i$
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  + (0.982 − 0.186i)3-s + (0.787 + 0.615i)5-s + (0.877 + 0.479i)7-s + (0.930 − 0.366i)9-s + (0.915 − 0.401i)11-s + (−0.838 − 0.544i)13-s + (0.889 + 0.457i)15-s + (−0.441 − 0.897i)17-s + (0.990 + 0.137i)19-s + (0.951 + 0.307i)21-s + (−0.253 − 0.967i)23-s + (0.241 + 0.970i)25-s + (0.845 − 0.533i)27-s + (0.0312 − 0.999i)29-s + (0.143 − 0.989i)31-s + ⋯
L(s)  = 1  + (0.982 − 0.186i)3-s + (0.787 + 0.615i)5-s + (0.877 + 0.479i)7-s + (0.930 − 0.366i)9-s + (0.915 − 0.401i)11-s + (−0.838 − 0.544i)13-s + (0.889 + 0.457i)15-s + (−0.441 − 0.897i)17-s + (0.990 + 0.137i)19-s + (0.951 + 0.307i)21-s + (−0.253 − 0.967i)23-s + (0.241 + 0.970i)25-s + (0.845 − 0.533i)27-s + (0.0312 − 0.999i)29-s + (0.143 − 0.989i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.963 - 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.963 - 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4024\)    =    \(2^{3} \cdot 503\)
Sign: $0.963 - 0.265i$
Analytic conductor: \(18.6873\)
Root analytic conductor: \(18.6873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4024} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4024,\ (0:\ ),\ 0.963 - 0.265i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.796323030 - 0.5139780043i\)
\(L(\frac12)\) \(\approx\) \(3.796323030 - 0.5139780043i\)
\(L(1)\) \(\approx\) \(1.996087639 - 0.07654192658i\)
\(L(1)\) \(\approx\) \(1.996087639 - 0.07654192658i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
503 \( 1 \)
good3 \( 1 + (0.982 - 0.186i)T \)
5 \( 1 + (0.787 + 0.615i)T \)
7 \( 1 + (0.877 + 0.479i)T \)
11 \( 1 + (0.915 - 0.401i)T \)
13 \( 1 + (-0.838 - 0.544i)T \)
17 \( 1 + (-0.441 - 0.897i)T \)
19 \( 1 + (0.990 + 0.137i)T \)
23 \( 1 + (-0.253 - 0.967i)T \)
29 \( 1 + (0.0312 - 0.999i)T \)
31 \( 1 + (0.143 - 0.989i)T \)
37 \( 1 + (0.485 + 0.874i)T \)
41 \( 1 + (-0.372 + 0.928i)T \)
43 \( 1 + (-0.980 + 0.198i)T \)
47 \( 1 + (0.984 - 0.174i)T \)
53 \( 1 + (0.990 - 0.137i)T \)
59 \( 1 + (-0.265 + 0.964i)T \)
61 \( 1 + (-0.824 - 0.565i)T \)
67 \( 1 + (-0.889 + 0.457i)T \)
71 \( 1 + (-0.205 - 0.978i)T \)
73 \( 1 + (0.810 - 0.585i)T \)
79 \( 1 + (-0.965 + 0.259i)T \)
83 \( 1 + (-0.360 + 0.932i)T \)
89 \( 1 + (0.988 - 0.149i)T \)
97 \( 1 + (-0.528 - 0.848i)T \)
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

−18.43913607492788563234550990269, −17.67919354242493627392317860240, −17.237466638354660803868766943374, −16.53538240504638827433665637123, −15.73777209057846663885948444117, −14.87410589433921288932805472890, −14.32300489007931618872685589679, −13.89301877971214222341171818222, −13.2491112672946300177042244511, −12.3572559178098516594850303563, −11.80889240045786052617544559800, −10.691120797489934163463418015935, −10.11253680205809293044004692507, −9.282247007973349553238321859557, −8.96548869454458102610945319806, −8.16195527122311505649928008119, −7.28341156965381715250334895744, −6.84749059678506062505728097071, −5.59638194291933016902906881526, −4.88762836740963280709609777408, −4.24671287732584295749048564713, −3.55875759569942587803649415366, −2.409854611788424301535930777949, −1.64118824631746527914040848853, −1.296282949328266521747558825717, 0.966496596113133957618726667709, 1.842759633162513946250584016638, 2.619803293062985935164295017214, 2.9945909183165030132019310219, 4.16439870716632055693563209996, 4.889590127619386663384632982853, 5.823758083459158469602749984133, 6.5561610183754607803890741617, 7.3436717954342696450804016835, 7.980935565749060047089084675242, 8.72338963910065545561305853193, 9.524497223248110985505081881699, 9.85766345294319933568755521183, 10.813583154311788885479091143070, 11.7555454592239971655569654099, 12.11471628483479651111685482178, 13.39942480202982094544693035521, 13.65176578502452217394983427480, 14.39919962258709214173743677773, 14.95254509933444947091576322938, 15.31809540796058793446599289083, 16.4952314292607082057194959994, 17.17869386811830436410886477978, 18.04093589102517859971408625666, 18.37644970902676217198763842115

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