↑

Properties

Label 1-4033-4033.63-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.941 + 0.337i$
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  + (−0.5 + 0.866i)2-s + 3-s + (−0.5 − 0.866i)4-s + 5-s + (−0.5 + 0.866i)6-s + 7-s + 8-s + 9-s + (−0.5 + 0.866i)10-s + (−0.5 − 0.866i)11-s + (−0.5 − 0.866i)12-s + 13-s + (−0.5 + 0.866i)14-s + 15-s + (−0.5 + 0.866i)16-s + (−0.5 + 0.866i)17-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + 3-s + (−0.5 − 0.866i)4-s + 5-s + (−0.5 + 0.866i)6-s + 7-s + 8-s + 9-s + (−0.5 + 0.866i)10-s + (−0.5 − 0.866i)11-s + (−0.5 − 0.866i)12-s + 13-s + (−0.5 + 0.866i)14-s + 15-s + (−0.5 + 0.866i)16-s + (−0.5 + 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.941 + 0.337i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.941 + 0.337i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.941 + 0.337i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (63, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.941 + 0.337i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.911861808 + 0.5066184042i\)
\(L(\frac12)\) \(\approx\) \(2.911861808 + 0.5066184042i\)
\(L(1)\) \(\approx\) \(1.543621166 + 0.3752267778i\)
\(L(1)\) \(\approx\) \(1.543621166 + 0.3752267778i\)

Euler product

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

−18.545234170055568747921066462430, −17.85564267202395108348398156908, −17.48364835117123089222897103825, −16.51851800821127488017705469277, −15.7155367316536844361991659835, −14.79616698116939588913788782712, −14.22351525830875107338445189912, −13.54191640224076787673026305965, −13.00947193196501363785255674295, −12.43344426475002538329142867099, −11.32939654560291646904196641734, −10.76545248423931521742009264630, −10.054818953321655500702564001724, −9.443062385958931318197311171498, −8.79598528960837730751625010700, −8.238912107863745797039174996052, −7.478346041562974640588158753, −6.78498999518965888616656685828, −5.506186016025677364853759660050, −4.65787519024594204816142874363, −4.0977470050563780135808274866, −2.99390829267781090070641524987, −2.4193612468095110371625823846, −1.65999705494042544888434274583, −1.22560610164551144213788099344, 0.906071768715870204046329514868, 1.71768092864884184585144325012, 2.316417307161099421481801617371, 3.48140982365279198516801700011, 4.39485938487426016636430785800, 5.22315310777613647459789319876, 5.840487366647524019443680955921, 6.71521582728045492453841025470, 7.35213041941965205797767083892, 8.32116391656935440365721526626, 8.72510263686159058553882363353, 9.060626690349431791446156250182, 10.10799000714027450083246769672, 10.801308465417341990120118205115, 11.164971798983258745463782403555, 12.96252120642441840728763832432, 13.25143516783015814900172678334, 13.799064619636921050374790133582, 14.67910219480271855431904012039, 14.88480778317755871339959174061, 15.767260102086112798133752381999, 16.42441052524704755663048997636, 17.30491132204698266028676935675, 17.75000213256053789356962197186, 18.537546670076759135175312870902

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