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Properties

Label 1-4033-4033.1049-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.504 + 0.863i$
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 + (−0.342 − 0.939i)5-s + (0.5 + 0.866i)6-s + (−0.939 + 0.342i)7-s + (−0.5 − 0.866i)8-s + (0.173 + 0.984i)9-s + i·10-s + i·11-s + (−0.173 − 0.984i)12-s + (0.173 − 0.984i)13-s + 14-s + (−0.342 + 0.939i)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 + (−0.342 − 0.939i)5-s + (0.5 + 0.866i)6-s + (−0.939 + 0.342i)7-s + (−0.5 − 0.866i)8-s + (0.173 + 0.984i)9-s + i·10-s + i·11-s + (−0.173 − 0.984i)12-s + (0.173 − 0.984i)13-s + 14-s + (−0.342 + 0.939i)15-s + (0.173 + 0.984i)16-s + (−0.766 + 0.642i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.504 + 0.863i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.504 + 0.863i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.504 + 0.863i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (1049, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.504 + 0.863i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.06265205661 - 0.1092119955i\)
\(L(\frac12)\) \(\approx\) \(-0.06265205661 - 0.1092119955i\)
\(L(1)\) \(\approx\) \(0.3561277882 - 0.1834316733i\)
\(L(1)\) \(\approx\) \(0.3561277882 - 0.1834316733i\)

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

−18.7454425506527642742128722036, −18.28511518024028014597130364019, −17.50850437834389000228365039064, −16.73467655734286947922493903177, −16.304031644667229261428668574658, −15.773307517936009170854561463868, −15.12749653728254922766092537208, −14.4145129241490061544758411091, −13.61237509118214456919357311159, −12.62398621426149037860591084430, −11.53035075205044603274467375628, −11.123805954417661466246255439004, −10.80362918705726386645294231474, −9.912629667463708669013874463182, −9.23490563540413551822218381521, −8.816285919912907306283737368983, −7.54303153070000112108124700582, −6.92135543985831671071267633238, −6.396154252873921125262442113814, −5.890811935460384937081816553, −4.83766810524378019928375399545, −3.80370325154033695410259000114, −3.20472617666351396728765453729, −2.24102634371586386893066577154, −0.93558554681515302995821931607, 0.08949504375401059912412920972, 0.76326327890671073320610869086, 1.96848435858002031264020899772, 2.318170525325384014329365429070, 3.68148042829822656233570524688, 4.36788497882745801285238162158, 5.52526784066495539270274248065, 6.092271362730833220226191706536, 6.9521567836898664570973389771, 7.528164874984117716806385264946, 8.34368528405753301520810348206, 8.92771977848995628919028309894, 9.69501362884431128966358539790, 10.56943715694300693373694065568, 10.93803931136438153881268491987, 12.07905807326802170131075312465, 12.41752288826826374056567282413, 12.90095558151929068675578907735, 13.34854088780049440708264079948, 15.11881054045805927869087625832, 15.42103925699347675515050652959, 16.25696928333123509623946549592, 16.98133117419692719244738453165, 17.17869355738002672063413052085, 18.042877714082743646684913130487

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