↑

Properties

Label 1-4033-4033.1523-r1-0-0
Degree $1$
Conductor $4033$
Sign $0.327 + 0.944i$
Analytic cond. $433.406$
Root an. cond. $433.406$
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  + (−0.342 − 0.939i)2-s + (−0.973 − 0.230i)3-s + (−0.766 + 0.642i)4-s + (0.549 − 0.835i)5-s + (0.116 + 0.993i)6-s + (−0.0581 + 0.998i)7-s + (0.866 + 0.5i)8-s + (0.893 + 0.448i)9-s + (−0.973 − 0.230i)10-s + (0.973 − 0.230i)11-s + (0.893 − 0.448i)12-s + (0.998 + 0.0581i)13-s + (0.957 − 0.286i)14-s + (−0.727 + 0.686i)15-s + (0.173 − 0.984i)16-s + (−0.984 − 0.173i)17-s + ⋯
L(s)  = 1  + (−0.342 − 0.939i)2-s + (−0.973 − 0.230i)3-s + (−0.766 + 0.642i)4-s + (0.549 − 0.835i)5-s + (0.116 + 0.993i)6-s + (−0.0581 + 0.998i)7-s + (0.866 + 0.5i)8-s + (0.893 + 0.448i)9-s + (−0.973 − 0.230i)10-s + (0.973 − 0.230i)11-s + (0.893 − 0.448i)12-s + (0.998 + 0.0581i)13-s + (0.957 − 0.286i)14-s + (−0.727 + 0.686i)15-s + (0.173 − 0.984i)16-s + (−0.984 − 0.173i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.327 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.327 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.327 + 0.944i$
Analytic conductor: \(433.406\)
Root analytic conductor: \(433.406\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (1523, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (1:\ ),\ 0.327 + 0.944i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02935893488 + 0.02090498620i\)
\(L(\frac12)\) \(\approx\) \(0.02935893488 + 0.02090498620i\)
\(L(1)\) \(\approx\) \(0.5547377047 - 0.3767731830i\)
\(L(1)\) \(\approx\) \(0.5547377047 - 0.3767731830i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (-0.342 - 0.939i)T \)
3 \( 1 + (-0.973 - 0.230i)T \)
5 \( 1 + (0.549 - 0.835i)T \)
7 \( 1 + (-0.0581 + 0.998i)T \)
11 \( 1 + (0.973 - 0.230i)T \)
13 \( 1 + (0.998 + 0.0581i)T \)
17 \( 1 + (-0.984 - 0.173i)T \)
19 \( 1 + (0.642 - 0.766i)T \)
23 \( 1 + (-0.642 - 0.766i)T \)
29 \( 1 + (0.727 + 0.686i)T \)
31 \( 1 + (-0.998 + 0.0581i)T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (-0.342 - 0.939i)T \)
47 \( 1 + (-0.396 - 0.918i)T \)
53 \( 1 + (0.835 + 0.549i)T \)
59 \( 1 + (-0.727 + 0.686i)T \)
61 \( 1 + (-0.918 - 0.396i)T \)
67 \( 1 + (-0.0581 - 0.998i)T \)
71 \( 1 + (-0.939 - 0.342i)T \)
73 \( 1 + (-0.973 + 0.230i)T \)
79 \( 1 + (0.448 - 0.893i)T \)
83 \( 1 + (-0.286 - 0.957i)T \)
89 \( 1 + (0.116 + 0.993i)T \)
97 \( 1 + (0.448 + 0.893i)T \)
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

−18.33795133727292903256790996536, −18.04231412985244391993583479446, −17.42993095484624906290275327518, −16.90291442416881662091163923445, −16.19921254512479111990936195497, −15.6606680604879718166728953250, −14.80474043746302989991872256094, −14.2113007643595671488535396068, −13.49025053187797112162387453091, −13.01489084202322639453468142524, −11.714309348082538840780058965223, −11.12124720332859979248815403399, −10.470719989017260255696424555289, −9.811950508799922514615464907526, −9.402664030449716931361188599974, −8.27110718012116604245551482821, −7.36096920705207450330999990438, −6.85131368183287661149039466695, −6.12278663793366639322806286851, −5.89903401058682220311614661643, −4.71919946740675107044021577978, −4.048849432987901623173093595410, −3.42768994802303697543409824280, −1.60157507142197342641110259788, −1.23965322186314430366557312311, 0.008910693167983868014809809038, 0.78071438370220886906781197400, 1.639123945196392903224378048163, 2.111459589663522109035617370227, 3.28793771490951720650991259602, 4.27146512850519952641272853210, 4.89304381485023710698631852200, 5.63996702577119663576710111214, 6.3256823292152747876552779821, 7.14836605795877447755614589102, 8.39774032366183524461980387037, 8.92322847317218381822642531810, 9.28551839724689278105535304136, 10.32121781127992266020701960722, 10.90705242501226203796010020195, 11.77142331230203076612091540300, 12.05138816337474053645443913652, 12.71451979836427533587107473508, 13.52970223759231254368931941173, 13.83527280840122334769499221507, 15.162409964578624519613891942227, 16.20314033409688836013775133545, 16.38177473814891398912794726595, 17.336528569564304636070361345692, 17.865778847133333129969089051318

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