↑

Properties

Label 1-4033-4033.3325-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.994 - 0.102i$
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  + (0.939 + 0.342i)2-s + (0.5 + 0.866i)3-s + (0.766 + 0.642i)4-s + (0.866 + 0.5i)5-s + (0.173 + 0.984i)6-s + (−0.5 + 0.866i)7-s + (0.5 + 0.866i)8-s + (−0.5 + 0.866i)9-s + (0.642 + 0.766i)10-s + (−0.642 + 0.766i)11-s + (−0.173 + 0.984i)12-s + (0.5 + 0.866i)13-s + (−0.766 + 0.642i)14-s + i·15-s + (0.173 + 0.984i)16-s + (0.766 − 0.642i)17-s + ⋯
L(s)  = 1  + (0.939 + 0.342i)2-s + (0.5 + 0.866i)3-s + (0.766 + 0.642i)4-s + (0.866 + 0.5i)5-s + (0.173 + 0.984i)6-s + (−0.5 + 0.866i)7-s + (0.5 + 0.866i)8-s + (−0.5 + 0.866i)9-s + (0.642 + 0.766i)10-s + (−0.642 + 0.766i)11-s + (−0.173 + 0.984i)12-s + (0.5 + 0.866i)13-s + (−0.766 + 0.642i)14-s + i·15-s + (0.173 + 0.984i)16-s + (0.766 − 0.642i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.102i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.102i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.994 - 0.102i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (3325, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.994 - 0.102i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2301660164 + 4.488251686i\)
\(L(\frac12)\) \(\approx\) \(-0.2301660164 + 4.488251686i\)
\(L(1)\) \(\approx\) \(1.480217868 + 1.894529068i\)
\(L(1)\) \(\approx\) \(1.480217868 + 1.894529068i\)

Euler product

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

−18.160866539754941979224713318172, −17.49255681092374977721301585986, −16.764255443170890911038463681767, −15.902327737183260429434682106823, −15.34981599354043186431582994434, −14.23940122483923437503499838691, −13.863053980899600901844590531934, −13.29298391261681035788924299019, −12.924396852242902296045962221763, −12.290367327043167944147812674103, −11.37747421445199373021113050497, −10.42781425041274941017718947643, −10.12249027711230876332932899335, −9.09314323539743888390070712973, −8.23462908510509367050329396290, −7.562716482109844361016599084797, −6.6087465354140426902628181239, −6.11430950716540433781523679788, −5.47249469476004089059574100609, −4.61444661512311488948767126365, −3.43719449507860225215460718620, −3.12015481051195982435482706299, −2.241547144578751890124364864734, −1.21021489262249860783322281987, −0.80890794540210789101510057686, 1.94970683534804544062947111908, 2.313622119199942833501891829479, 3.109776674016998835384935262988, 3.756696558413224540653209716958, 4.78981878399203831218614923274, 5.21467724949717438263672407775, 6.21089513141786417128821575599, 6.46820578400833245178266614945, 7.68590265575335420382818142675, 8.28711751921091632279441953128, 9.24000984482971763085183414718, 9.88483899952516018621631098225, 10.43692472153529250755371833138, 11.303972051256141783535357612080, 12.16270592948350921372451417533, 12.72027694800480626703536940620, 13.703672474087400478380414814951, 14.0833204691103254829189392513, 14.665869001666735015944244322442, 15.40750004919626090290533482746, 15.869505722445724414460087595531, 16.569036599241437253727444640517, 17.14346330355584810865005378587, 18.25288027981544166619379408783, 18.7316578219439756684635038613

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