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Properties

Label 1-4033-4033.1236-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.330 - 0.943i$
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  + 2-s + (0.286 − 0.957i)3-s + 4-s + (0.448 − 0.893i)5-s + (0.286 − 0.957i)6-s + (−0.835 + 0.549i)7-s + 8-s + (−0.835 − 0.549i)9-s + (0.448 − 0.893i)10-s + (−0.448 − 0.893i)11-s + (0.286 − 0.957i)12-s + (0.893 + 0.448i)13-s + (−0.835 + 0.549i)14-s + (−0.727 − 0.686i)15-s + 16-s + (0.5 − 0.866i)17-s + ⋯
L(s)  = 1  + 2-s + (0.286 − 0.957i)3-s + 4-s + (0.448 − 0.893i)5-s + (0.286 − 0.957i)6-s + (−0.835 + 0.549i)7-s + 8-s + (−0.835 − 0.549i)9-s + (0.448 − 0.893i)10-s + (−0.448 − 0.893i)11-s + (0.286 − 0.957i)12-s + (0.893 + 0.448i)13-s + (−0.835 + 0.549i)14-s + (−0.727 − 0.686i)15-s + 16-s + (0.5 − 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.330 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.330 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.330 - 0.943i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (1236, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.330 - 0.943i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.500732926 - 3.524363993i\)
\(L(\frac12)\) \(\approx\) \(2.500732926 - 3.524363993i\)
\(L(1)\) \(\approx\) \(2.006709617 - 1.149937386i\)
\(L(1)\) \(\approx\) \(2.006709617 - 1.149937386i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + T \)
3 \( 1 + (0.286 - 0.957i)T \)
5 \( 1 + (0.448 - 0.893i)T \)
7 \( 1 + (-0.835 + 0.549i)T \)
11 \( 1 + (-0.448 - 0.893i)T \)
13 \( 1 + (0.893 + 0.448i)T \)
17 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (0.766 + 0.642i)T \)
23 \( 1 + (0.939 - 0.342i)T \)
29 \( 1 + (0.549 + 0.835i)T \)
31 \( 1 + (0.802 - 0.597i)T \)
41 \( 1 + (0.342 - 0.939i)T \)
43 \( 1 + (0.642 + 0.766i)T \)
47 \( 1 + (-0.957 + 0.286i)T \)
53 \( 1 + (-0.230 - 0.973i)T \)
59 \( 1 + (-0.686 + 0.727i)T \)
61 \( 1 + (0.918 - 0.396i)T \)
67 \( 1 + (-0.957 + 0.286i)T \)
71 \( 1 - T \)
73 \( 1 + (0.835 + 0.549i)T \)
79 \( 1 + (-0.686 + 0.727i)T \)
83 \( 1 + (0.993 + 0.116i)T \)
89 \( 1 + (0.998 - 0.0581i)T \)
97 \( 1 + (-0.802 - 0.597i)T \)
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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

−18.99792213375964228848681106765, −17.72897903126982859117735476014, −17.2600250126615808168808191053, −16.323293959854008004018273681968, −15.72799802805255710833172863027, −15.21081711276633117902645525536, −14.7143051983529934835423928054, −13.761109234084451813974350792848, −13.5025255768593146000427749960, −12.74991629587540902590439874220, −11.771156216690376686600054377298, −10.92851766155163905738047174493, −10.44742859735144882594014138584, −9.99136035514553601426993505104, −9.2380093685493756960132755469, −7.99994913966964035159206986628, −7.37250235790590364585249857123, −6.49800831794206856111787856332, −5.95725828604509917416076258796, −5.13112991858972904893161672510, −4.37857312806813473143139853113, −3.5169684487877120510971295973, −3.08855178056195256185299098183, −2.47174098391424845639284036278, −1.284787739698032105496747519904, 0.83436758015466900051896470416, 1.44356237893022029755730900174, 2.56040150091264691750376717168, 3.02279114439540160291421028461, 3.79981737035087050009936345426, 5.00706219078596748607241066133, 5.61190013392453434826205747618, 6.165078192421668446640176255, 6.76999187670916159253261856662, 7.71856108959464987735038587981, 8.44139757943086812323292230241, 9.10607053461138234122799727367, 9.905165953806800967459714009474, 10.99367752392694283164542311632, 11.759887198643667007466753980523, 12.26779981595385139323524471278, 12.96960355548428576982399815059, 13.38247747069114773104927489261, 13.955551899863522938236422068262, 14.530905568583543103593726316050, 15.65229981263229503029657254082, 16.28229371174684600145217399331, 16.50378262442440836556796608567, 17.5994447928863174062428711724, 18.46604442447309897324844211677

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