↑

Properties

Label 1-6027-6027.632-r0-0-0
Degree $1$
Conductor $6027$
Sign $-0.187 + 0.982i$
Analytic cond. $27.9892$
Root an. cond. $27.9892$
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  + (0.817 − 0.575i)2-s + (0.337 − 0.941i)4-s + (0.460 + 0.887i)5-s + (−0.266 − 0.963i)8-s + (0.887 + 0.460i)10-s + (−0.830 − 0.557i)11-s + (0.640 − 0.767i)13-s + (−0.772 − 0.635i)16-s + (−0.996 + 0.0821i)17-s + (0.838 + 0.544i)19-s + (0.990 − 0.134i)20-s + (−0.999 + 0.0224i)22-s + (−0.925 + 0.379i)23-s + (−0.575 + 0.817i)25-s + (0.0821 − 0.996i)26-s + ⋯
L(s)  = 1  + (0.817 − 0.575i)2-s + (0.337 − 0.941i)4-s + (0.460 + 0.887i)5-s + (−0.266 − 0.963i)8-s + (0.887 + 0.460i)10-s + (−0.830 − 0.557i)11-s + (0.640 − 0.767i)13-s + (−0.772 − 0.635i)16-s + (−0.996 + 0.0821i)17-s + (0.838 + 0.544i)19-s + (0.990 − 0.134i)20-s + (−0.999 + 0.0224i)22-s + (−0.925 + 0.379i)23-s + (−0.575 + 0.817i)25-s + (0.0821 − 0.996i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.187 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.187 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6027\)    =    \(3 \cdot 7^{2} \cdot 41\)
Sign: $-0.187 + 0.982i$
Analytic conductor: \(27.9892\)
Root analytic conductor: \(27.9892\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6027} (632, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6027,\ (0:\ ),\ -0.187 + 0.982i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4243629506 + 0.5129660529i\)
\(L(\frac12)\) \(\approx\) \(0.4243629506 + 0.5129660529i\)
\(L(1)\) \(\approx\) \(1.308849917 - 0.3548506575i\)
\(L(1)\) \(\approx\) \(1.308849917 - 0.3548506575i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (0.817 - 0.575i)T \)
5 \( 1 + (0.460 + 0.887i)T \)
11 \( 1 + (-0.830 - 0.557i)T \)
13 \( 1 + (0.640 - 0.767i)T \)
17 \( 1 + (-0.996 + 0.0821i)T \)
19 \( 1 + (0.838 + 0.544i)T \)
23 \( 1 + (-0.925 + 0.379i)T \)
29 \( 1 + (-0.287 - 0.957i)T \)
31 \( 1 + (-0.913 + 0.406i)T \)
37 \( 1 + (-0.999 + 0.0299i)T \)
43 \( 1 + (-0.834 - 0.550i)T \)
47 \( 1 + (-0.171 + 0.985i)T \)
53 \( 1 + (-0.427 - 0.904i)T \)
59 \( 1 + (0.447 + 0.894i)T \)
61 \( 1 + (0.611 + 0.791i)T \)
67 \( 1 + (-0.629 + 0.777i)T \)
71 \( 1 + (0.957 + 0.287i)T \)
73 \( 1 + (-0.294 + 0.955i)T \)
79 \( 1 + (-0.965 - 0.258i)T \)
83 \( 1 + (0.222 + 0.974i)T \)
89 \( 1 + (0.938 + 0.344i)T \)
97 \( 1 + (-0.987 - 0.156i)T \)
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

−17.445318193427828845232453695921, −16.63823417550084238457079832682, −16.11662554155587071287717372390, −15.69568751680060625205773277046, −14.95735668564380816785989566454, −14.0277848424429106488820587378, −13.667951068596144123409480101759, −12.97752988706755223865182048507, −12.550596794191772319292143001649, −11.73651271573772924261245008221, −11.13604918272050315153570162165, −10.23994059133696586018052783024, −9.29384100703848261624756513855, −8.800801370609291293051766709473, −8.09694047524539672433821919624, −7.31709361750304753155962115630, −6.63647212659749696982707952911, −5.95021827622178254475556044224, −5.11073797250185947141049952998, −4.797216839452495320603294584854, −3.984821205902700058159663637114, −3.19166401202445359079075227363, −2.09934996014370735188800455929, −1.73359935395417756118782844041, −0.106007018688927892646158080251, 1.223489913306109241406078728719, 2.08547039617112042531778864790, 2.69936976962011595023009606645, 3.488722796318494007446735799167, 3.90169361577701803476849026570, 5.13683844858056411160133979599, 5.64018906090806848045352919476, 6.15818560353870163343666597872, 6.96153832738869230621866926506, 7.71354885761513844393003880310, 8.58159105802566302185990045925, 9.54281678852834093210842042295, 10.2585871469928056539087714468, 10.5851946647776731264395622305, 11.36556180257544615743109473295, 11.78913571993231744500467925408, 12.92715280563443683168546120842, 13.25328453084165522776661323747, 13.90735207439081634470107115354, 14.39950227599162597935891729825, 15.271132325418488921825712808105, 15.711529529285856575982298879698, 16.2714075112458673292023935405, 17.504014077208529643692658767817, 18.09330503192730387066804880271

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