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Properties

Label 1-4033-4033.708-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.992 - 0.125i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.992 - 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.992 - 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.992 - 0.125i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (708, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.992 - 0.125i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1604715607 + 2.553732910i\)
\(L(\frac12)\) \(\approx\) \(-0.1604715607 + 2.553732910i\)
\(L(1)\) \(\approx\) \(1.295528196 + 1.155822958i\)
\(L(1)\) \(\approx\) \(1.295528196 + 1.155822958i\)

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

−18.37385551550049199866354139255, −17.45675104899919557546074274598, −16.74956268063740053459170685309, −15.70595324913964890580392595911, −15.188946137103308574360717704585, −14.64900378762220439197630359017, −13.93959579733947592833967388803, −13.287714833113039912404353459022, −12.675824830123941070873653318871, −12.194404236702141544394451948484, −11.30178278970773559903798809101, −10.85066756880489495161095951189, −9.90580601185012538139101774622, −9.22275521866472558767649195775, −7.82056098234818076411129035016, −7.63797399751410583655312441239, −6.80636496469485844148609193054, −6.303377554918206005315930505725, −5.41295084651238931337876359038, −4.16295482485846309606490958939, −3.69478475842285523904069269027, −3.20872088748398199485440767091, −2.24457781213070925981948383312, −1.372070892462067235011551966298, −0.45114851721037416732607001015, 1.48619782530988339933644950551, 2.546402570447804077165497120195, 3.38332102986899593790642532983, 3.81536892491080449818562196442, 4.44463780022080563031089400925, 5.39576633547462939957592067370, 5.864606402572546614308629043033, 6.79687743607983188930404012029, 7.73408656046868657063840258738, 8.46715022315391379627499012398, 8.90234669719012983935074876314, 9.67509711802055680612583174069, 10.83451863657397959830410768857, 11.37884568126856350036952113242, 12.14379315865538572479136718586, 12.51720466946927998894016930997, 13.606094732600552100047632884791, 14.14730392188259056565120997439, 14.82030410032111056626598352418, 15.33455463128225160559063421656, 16.20840704578374738693677209098, 16.56542411816473168489971550162, 16.63640400908503643551921789184, 18.35573448355898521959016390699, 18.97720176467609717228868496648

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