Properties

Label 1-4033-4033.2945-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

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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: $\chi_{4033} (2945, \cdot )$
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 \( 1 + (0.984 + 0.173i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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