↑

Properties

Label 1-4033-4033.284-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.779 - 0.626i$
Analytic cond. $18.7291$
Root an. cond. $18.7291$
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  + (−0.766 − 0.642i)2-s + (−0.5 − 0.866i)3-s + (0.173 + 0.984i)4-s − 5-s + (−0.173 + 0.984i)6-s + 7-s + (0.5 − 0.866i)8-s + (−0.5 + 0.866i)9-s + (0.766 + 0.642i)10-s + (0.766 − 0.642i)11-s + (0.766 − 0.642i)12-s + (0.5 + 0.866i)13-s + (−0.766 − 0.642i)14-s + (0.5 + 0.866i)15-s + (−0.939 + 0.342i)16-s + (−0.173 + 0.984i)17-s + ⋯
L(s)  = 1  + (−0.766 − 0.642i)2-s + (−0.5 − 0.866i)3-s + (0.173 + 0.984i)4-s − 5-s + (−0.173 + 0.984i)6-s + 7-s + (0.5 − 0.866i)8-s + (−0.5 + 0.866i)9-s + (0.766 + 0.642i)10-s + (0.766 − 0.642i)11-s + (0.766 − 0.642i)12-s + (0.5 + 0.866i)13-s + (−0.766 − 0.642i)14-s + (0.5 + 0.866i)15-s + (−0.939 + 0.342i)16-s + (−0.173 + 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.779 - 0.626i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.779 - 0.626i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.779 - 0.626i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (284, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.779 - 0.626i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9194585355 - 0.3235177817i\)
\(L(\frac12)\) \(\approx\) \(0.9194585355 - 0.3235177817i\)
\(L(1)\) \(\approx\) \(0.6266594378 - 0.2575481108i\)
\(L(1)\) \(\approx\) \(0.6266594378 - 0.2575481108i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (-0.766 - 0.642i)T \)
3 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 - T \)
7 \( 1 + T \)
11 \( 1 + (0.766 - 0.642i)T \)
13 \( 1 + (0.5 + 0.866i)T \)
17 \( 1 + (-0.173 + 0.984i)T \)
19 \( 1 + (0.939 + 0.342i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (-0.173 + 0.984i)T \)
31 \( 1 + (-0.766 + 0.642i)T \)
41 \( 1 + (-0.939 - 0.342i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 + (0.766 - 0.642i)T \)
53 \( 1 + (0.173 + 0.984i)T \)
59 \( 1 + (0.5 + 0.866i)T \)
61 \( 1 + (0.5 - 0.866i)T \)
67 \( 1 + (0.173 - 0.984i)T \)
71 \( 1 + (0.766 - 0.642i)T \)
73 \( 1 + (-0.939 - 0.342i)T \)
79 \( 1 + (-0.173 + 0.984i)T \)
83 \( 1 + (0.766 - 0.642i)T \)
89 \( 1 + (-0.766 + 0.642i)T \)
97 \( 1 + (-0.173 + 0.984i)T \)
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

−18.24606243179473216485441586954, −17.7329266794473491897619756512, −17.24184014929751054769906007801, −16.4563455954945065509306876783, −15.76131683564381501979777681412, −15.38336832376153032705369989017, −14.77247407988104766666671358774, −14.21927979919620729565101610238, −13.13921156275225279186494442011, −11.83608567112820538001527874857, −11.461601349153100694417537887696, −11.105229432280647876132138036216, −10.1433742432501861514974099487, −9.4661914248525286276361323764, −8.866532364928417133255933601702, −8.05541481303867169514807155576, −7.42484419353540520042578558657, −6.81054071099988915979732051460, −5.69844402360053627628417961101, −5.16422950666757899084201446189, −4.48178663721692626136359622517, −3.743780581026039772239853019681, −2.675010965565443711933592185309, −1.30785515169585182971054045154, −0.6045142041278770329665730423, 0.79839547998642833381636125076, 1.41125823383287158005919466203, 2.088821917602379397758046023067, 3.29376368965617878498367290997, 3.92382417441339512152171050161, 4.7731708987936257336998490109, 5.783993326227521536823302266610, 6.90864511962597731962063190225, 7.141396850216487189804879909905, 8.12379115187553835689977111595, 8.58832029745185503787538563152, 9.05117303805243779306401308674, 10.56441543301555169476191681379, 10.951412723008506835596639671870, 11.436267877485883994837464910463, 12.25643485740656654949085715544, 12.34029119870103186530256143943, 13.56820853942988476326432927986, 14.12799610081117866263837915999, 14.992585968910224739448945027951, 16.02944664130429660846034294549, 16.701906412003799977158531143139, 16.98881075586582636711700715042, 17.94510274344839575226038609552, 18.5431617398615191014317598717

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