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Properties

Label 1-4033-4033.470-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.855 - 0.518i$
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 + (0.766 − 0.642i)3-s + 4-s + (0.173 + 0.984i)5-s + (−0.766 + 0.642i)6-s + (0.173 + 0.984i)7-s − 8-s + (0.173 − 0.984i)9-s + (−0.173 − 0.984i)10-s + (−0.173 + 0.984i)11-s + (0.766 − 0.642i)12-s + (−0.173 − 0.984i)13-s + (−0.173 − 0.984i)14-s + (0.766 + 0.642i)15-s + 16-s − 17-s + ⋯
L(s)  = 1  − 2-s + (0.766 − 0.642i)3-s + 4-s + (0.173 + 0.984i)5-s + (−0.766 + 0.642i)6-s + (0.173 + 0.984i)7-s − 8-s + (0.173 − 0.984i)9-s + (−0.173 − 0.984i)10-s + (−0.173 + 0.984i)11-s + (0.766 − 0.642i)12-s + (−0.173 − 0.984i)13-s + (−0.173 − 0.984i)14-s + (0.766 + 0.642i)15-s + 16-s − 17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.855 - 0.518i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.855 - 0.518i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.855 - 0.518i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (470, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.855 - 0.518i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.06709638783 - 0.2400368185i\)
\(L(\frac12)\) \(\approx\) \(0.06709638783 - 0.2400368185i\)
\(L(1)\) \(\approx\) \(0.7518660698 + 0.008476882744i\)
\(L(1)\) \(\approx\) \(0.7518660698 + 0.008476882744i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 - T \)
3 \( 1 + (0.766 - 0.642i)T \)
5 \( 1 + (0.173 + 0.984i)T \)
7 \( 1 + (0.173 + 0.984i)T \)
11 \( 1 + (-0.173 + 0.984i)T \)
13 \( 1 + (-0.173 - 0.984i)T \)
17 \( 1 - T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (0.173 + 0.984i)T \)
31 \( 1 + (-0.939 + 0.342i)T \)
41 \( 1 + (0.5 + 0.866i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (-0.766 - 0.642i)T \)
53 \( 1 + (-0.766 + 0.642i)T \)
59 \( 1 + (-0.766 - 0.642i)T \)
61 \( 1 + (-0.939 - 0.342i)T \)
67 \( 1 + (-0.766 - 0.642i)T \)
71 \( 1 + T \)
73 \( 1 + (0.173 - 0.984i)T \)
79 \( 1 + (-0.766 - 0.642i)T \)
83 \( 1 + (-0.939 - 0.342i)T \)
89 \( 1 + (0.173 + 0.984i)T \)
97 \( 1 + (-0.939 - 0.342i)T \)
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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

−18.95991282961966596091570861149, −18.02374038234603722899472701939, −17.13228502502235380218740491096, −16.75280089806793089429421425517, −16.12998514743551048541314295221, −15.72019128663999521316464560468, −14.72897407942963770160504049039, −13.95693748557379943043622015165, −13.46693852646756614570699946737, −12.65140483642641054361256463779, −11.41024247295928114833534570338, −11.20252423803010730484477179008, −10.23306334950886821753981919325, −9.55743047269662618478518001741, −9.14914825142785046004994811890, −8.38792141770200244526878401338, −7.855246933115628792548208530675, −7.177585589812669963457692066222, −6.14440361600029597835851953165, −5.303849327460194548449894851397, −4.33965926449586676764317862783, −3.7548970475399142969952960106, −2.8158449203115663690958631209, −1.79139576889982559306738621106, −1.247919871706174010312892281413, 0.07907968417096242506571272254, 1.56458700113306085102341662366, 2.13252651279104795938036814058, 2.8919812536661275595811994229, 3.1915617176359393068497152463, 4.760087423052403659711062743841, 5.75183511885317876653280013855, 6.67723674360195797047577196385, 6.98973676593794759356368197097, 7.77286308565775341834448021843, 8.40998806663288770788716819778, 9.230928744831051809963181790265, 9.58560241667625355169006591223, 10.6269685250232869352316177735, 11.07016220946442607948695304232, 12.08104283381199197957522284516, 12.570466358283918412907754710530, 13.322246238854996452896259221992, 14.3887468027620824986302411925, 15.05509078353877609548865554781, 15.23290079846110885093544248606, 15.969627229092792234176886154798, 17.23473778007907781637226283366, 17.9405740921304139358570929927, 18.13726505269038810320342061415

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