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Properties

Label 1-4033-4033.2395-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.238 + 0.971i$
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  + (0.766 + 0.642i)2-s + (−0.686 + 0.727i)3-s + (0.173 + 0.984i)4-s + (−0.893 + 0.448i)5-s + (−0.993 + 0.116i)6-s + (−0.835 − 0.549i)7-s + (−0.5 + 0.866i)8-s + (−0.0581 − 0.998i)9-s + (−0.973 − 0.230i)10-s + (−0.973 + 0.230i)11-s + (−0.835 − 0.549i)12-s + (−0.835 − 0.549i)13-s + (−0.286 − 0.957i)14-s + (0.286 − 0.957i)15-s + (−0.939 + 0.342i)16-s + (−0.939 + 0.342i)17-s + ⋯
L(s)  = 1  + (0.766 + 0.642i)2-s + (−0.686 + 0.727i)3-s + (0.173 + 0.984i)4-s + (−0.893 + 0.448i)5-s + (−0.993 + 0.116i)6-s + (−0.835 − 0.549i)7-s + (−0.5 + 0.866i)8-s + (−0.0581 − 0.998i)9-s + (−0.973 − 0.230i)10-s + (−0.973 + 0.230i)11-s + (−0.835 − 0.549i)12-s + (−0.835 − 0.549i)13-s + (−0.286 − 0.957i)14-s + (0.286 − 0.957i)15-s + (−0.939 + 0.342i)16-s + (−0.939 + 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.238 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.238 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.238 + 0.971i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (2395, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.238 + 0.971i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3063690699 + 0.3905859769i\)
\(L(\frac12)\) \(\approx\) \(0.3063690699 + 0.3905859769i\)
\(L(1)\) \(\approx\) \(0.5496777532 + 0.4169261796i\)
\(L(1)\) \(\approx\) \(0.5496777532 + 0.4169261796i\)

Euler product

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

−18.65680069717829368646795084313, −17.70618897393334171555900952242, −16.812325645518157150345808468057, −16.100508220878850005197774069756, −15.52973313904100285485725903502, −14.99176226215171026789516811332, −13.843347935302353929006597384281, −13.211586440605185536849442749922, −12.64861199477147181187226702317, −12.32869663946319953972044794255, −11.419476212375610359064229129211, −11.11217633155759080048790153147, −10.24303927326904493155869290513, −9.30208183883262089496351804616, −8.61612830896480571544996322482, −7.538833477359925302271458491728, −6.89411878437525540694475927406, −6.23681443307178102352829372618, −5.36527793708108601089485010270, −4.795413786442531118669210800872, −4.15455190447064835477982838701, −2.88426169964299799746779949524, −2.545482932234369257223972761203, −1.47717779115050973469363315315, −0.33815155168614285932124355047, 0.327259487971291033994112928036, 2.44502397852632553085973294563, 3.11929883345889335107822622026, 3.83330735036526064402269750180, 4.557538779568699976644017596858, 5.00199959184632982840743323179, 6.030737488838927063095613997291, 6.73604159193881261066707474395, 7.12208225665935967330100175485, 8.10187099934953793110961570689, 8.73779231977316643917102474402, 9.910078611859170790372494027866, 10.60398805622581487131743792007, 10.9641685767724302758005293010, 12.04175024749961756318369147537, 12.51880419340709539173639312226, 13.06266662357433126223295359699, 14.023362185457680200582386561655, 14.94067229336828532840161343066, 15.34211217172582393839876732484, 15.74234380394231389776142848604, 16.543167966406118454542853028798, 17.015494513825883919141323276883, 17.73449621402319290816979979402, 18.478275619581300909140048364651

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