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Properties

Label 1-4033-4033.413-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.183 - 0.983i$
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  + (−0.5 − 0.866i)2-s + (−0.766 + 0.642i)3-s + (−0.5 + 0.866i)4-s + (−0.984 + 0.173i)5-s + (0.939 + 0.342i)6-s + (0.173 + 0.984i)7-s + 8-s + (0.173 − 0.984i)9-s + (0.642 + 0.766i)10-s + (−0.642 + 0.766i)11-s + (−0.173 − 0.984i)12-s + (0.173 + 0.984i)13-s + (0.766 − 0.642i)14-s + (0.642 − 0.766i)15-s + (−0.5 − 0.866i)16-s + (0.5 + 0.866i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.766 + 0.642i)3-s + (−0.5 + 0.866i)4-s + (−0.984 + 0.173i)5-s + (0.939 + 0.342i)6-s + (0.173 + 0.984i)7-s + 8-s + (0.173 − 0.984i)9-s + (0.642 + 0.766i)10-s + (−0.642 + 0.766i)11-s + (−0.173 − 0.984i)12-s + (0.173 + 0.984i)13-s + (0.766 − 0.642i)14-s + (0.642 − 0.766i)15-s + (−0.5 − 0.866i)16-s + (0.5 + 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.183 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.183 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.183 - 0.983i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (413, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.183 - 0.983i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1146280917 - 0.1380130881i\)
\(L(\frac12)\) \(\approx\) \(0.1146280917 - 0.1380130881i\)
\(L(1)\) \(\approx\) \(0.4759572424 + 0.03172435980i\)
\(L(1)\) \(\approx\) \(0.4759572424 + 0.03172435980i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 + (-0.766 + 0.642i)T \)
5 \( 1 + (-0.984 + 0.173i)T \)
7 \( 1 + (0.173 + 0.984i)T \)
11 \( 1 + (-0.642 + 0.766i)T \)
13 \( 1 + (0.173 + 0.984i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (0.642 + 0.766i)T \)
31 \( 1 + (0.984 + 0.173i)T \)
41 \( 1 - iT \)
43 \( 1 + (-0.866 + 0.5i)T \)
47 \( 1 + (-0.342 - 0.939i)T \)
53 \( 1 + (-0.984 + 0.173i)T \)
59 \( 1 + (0.766 + 0.642i)T \)
61 \( 1 + (0.342 - 0.939i)T \)
67 \( 1 + (0.984 + 0.173i)T \)
71 \( 1 + (0.5 - 0.866i)T \)
73 \( 1 + (-0.766 - 0.642i)T \)
79 \( 1 + (0.173 - 0.984i)T \)
83 \( 1 + (-0.766 + 0.642i)T \)
89 \( 1 + (-0.342 + 0.939i)T \)
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

−18.65386149435630357995789767399, −17.85749870599875706970709978460, −17.20495218504755087544334038324, −16.719616263450887179089662954561, −15.93522419347116727640689036190, −15.69297352900816998190541111552, −14.63075269216165436680387945220, −13.878026407750241421811567748588, −13.23870690825267177647751262160, −12.674600017589362016933826708599, −11.50095843886095122905643131577, −11.198725133444736215219871037677, −10.299247645826245652957653638176, −9.8738523826329719297899790410, −8.35966500791507561627562375686, −8.09118916062207178920958800285, −7.56561573153012790736740307850, −6.86986337014636829505664242979, −6.1032241447783505404264300025, −5.29716439217655025478928171265, −4.76711930896640917148507109321, −3.84290196856725968583994884372, −2.818299106647532032976938350873, −1.24810941128232051809843749433, −0.82192082371597326406580025973, 0.10245309479388920081991630739, 1.34610484193433479931697099398, 2.367580380167892897280121406194, 3.140045684236200304766425449034, 3.9945724680268520267585226984, 4.71173187090172494307691292599, 5.10758380865878021641090205007, 6.49024957695313065738984006716, 6.994164654979988251666480415181, 8.16573761424238328215718346167, 8.61942374543713924905772209422, 9.34367534356634384687338103901, 10.18484234302173316233158766935, 10.77512839353590398961795494872, 11.33090323890357238128773996332, 12.081627874519650268168644647230, 12.35681069516923581456330500498, 13.07634480406912359214431100007, 14.38500048187260220054163374834, 15.055265747000495326473308646737, 15.67290715398426468670509839412, 16.29556841267580182839376262464, 16.998662863215274945770104916630, 17.72714000023525338125944533707, 18.342925825613896410921671456750

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