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Properties

Label 1-4033-4033.19-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.139 + 0.990i$
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 + (−0.766 + 0.642i)3-s + (0.766 + 0.642i)4-s + (0.642 + 0.766i)5-s + (0.939 − 0.342i)6-s + (0.766 − 0.642i)7-s + (−0.5 − 0.866i)8-s + (0.173 − 0.984i)9-s + (−0.342 − 0.939i)10-s + (0.342 − 0.939i)11-s − 12-s + (0.173 + 0.984i)13-s + (−0.939 + 0.342i)14-s + (−0.984 − 0.173i)15-s + (0.173 + 0.984i)16-s + (−0.766 + 0.642i)17-s + ⋯
L(s)  = 1  + (−0.939 − 0.342i)2-s + (−0.766 + 0.642i)3-s + (0.766 + 0.642i)4-s + (0.642 + 0.766i)5-s + (0.939 − 0.342i)6-s + (0.766 − 0.642i)7-s + (−0.5 − 0.866i)8-s + (0.173 − 0.984i)9-s + (−0.342 − 0.939i)10-s + (0.342 − 0.939i)11-s − 12-s + (0.173 + 0.984i)13-s + (−0.939 + 0.342i)14-s + (−0.984 − 0.173i)15-s + (0.173 + 0.984i)16-s + (−0.766 + 0.642i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.139 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.139 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.139 + 0.990i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.139 + 0.990i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6410763780 + 0.7373718098i\)
\(L(\frac12)\) \(\approx\) \(0.6410763780 + 0.7373718098i\)
\(L(1)\) \(\approx\) \(0.6810256394 + 0.1804283696i\)
\(L(1)\) \(\approx\) \(0.6810256394 + 0.1804283696i\)

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 \( 1 + (-0.766 + 0.642i)T \)
5 \( 1 + (0.642 + 0.766i)T \)
7 \( 1 + (0.766 - 0.642i)T \)
11 \( 1 + (0.342 - 0.939i)T \)
13 \( 1 + (0.173 + 0.984i)T \)
17 \( 1 + (-0.766 + 0.642i)T \)
19 \( 1 + (0.766 + 0.642i)T \)
23 \( 1 - T \)
29 \( 1 + (0.984 + 0.173i)T \)
31 \( 1 + (-0.984 + 0.173i)T \)
41 \( 1 + (0.342 + 0.939i)T \)
43 \( 1 + (0.866 + 0.5i)T \)
47 \( 1 + (0.642 + 0.766i)T \)
53 \( 1 + (0.866 - 0.5i)T \)
59 \( 1 + (0.173 - 0.984i)T \)
61 \( 1 + (0.642 + 0.766i)T \)
67 \( 1 + (-0.866 - 0.5i)T \)
71 \( 1 + (0.939 - 0.342i)T \)
73 \( 1 + (-0.766 + 0.642i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (0.5 + 0.866i)T \)
89 \( 1 + (0.866 + 0.5i)T \)
97 \( 1 + (-0.642 + 0.766i)T \)
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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

−18.01474229378067922632146803345, −17.57847881795370779226971222361, −17.397771443907999647273276474029, −16.3075440810963180956861789240, −15.81714479833506599287188750205, −15.150963042536291418226318942097, −14.15140262433515429079694083807, −13.54221684217926255849849836690, −12.56186732124783624294939549862, −12.0188915164419066828273539862, −11.482881995243235969503111162687, −10.56299364566730385249344276331, −10.022143823952234128149693179805, −9.045328608668392864092732657259, −8.673049024302729387244724349574, −7.66206472579473728989447886609, −7.27378280192931703811755666390, −6.2954432161080038068832604390, −5.611643910879666928795538181486, −5.18622842276812344085917689332, −4.39817955697580223078983499232, −2.50066648400485769259331111242, −2.10508001186255641106572446318, −1.26062204342488736932576709662, −0.48488915871986310304282049400, 1.03632513123978546985449173212, 1.65274966644665712820673761624, 2.66894882792399997480896503310, 3.76595438949318066719379223166, 4.08310449428326089450315365439, 5.357733104789995825670408533356, 6.233859671389115531867664089, 6.598768190351499658353964423066, 7.48536353537488714197113053150, 8.35664405192829513209737636553, 9.15749212929257637773727779639, 9.76492721333873428230494437343, 10.52129295602110301423369212776, 10.937797901213786795540980380409, 11.466498617120205706976358268494, 12.05477822123734236814552724580, 13.13354302024355563611059100793, 14.11512311936187114994147050757, 14.48169825971439436538570269353, 15.49261396631439116697181010618, 16.35504976316709844366810570889, 16.58408785737335706429647422708, 17.49820661665929960022346994752, 17.87036406289517004612056163469, 18.397454465352284234263770702603

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