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Properties

Label 1-4033-4033.935-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.114 - 0.993i$
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 + (−0.5 − 0.866i)3-s + (−0.5 + 0.866i)4-s + (−0.5 − 0.866i)5-s + (−0.5 + 0.866i)6-s + (−0.5 − 0.866i)7-s + 8-s + (−0.5 + 0.866i)9-s + (−0.5 + 0.866i)10-s + (−0.5 − 0.866i)11-s + 12-s + (−0.5 − 0.866i)13-s + (−0.5 + 0.866i)14-s + (−0.5 + 0.866i)15-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.5 + 0.866i)4-s + (−0.5 − 0.866i)5-s + (−0.5 + 0.866i)6-s + (−0.5 − 0.866i)7-s + 8-s + (−0.5 + 0.866i)9-s + (−0.5 + 0.866i)10-s + (−0.5 − 0.866i)11-s + 12-s + (−0.5 − 0.866i)13-s + (−0.5 + 0.866i)14-s + (−0.5 + 0.866i)15-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.114 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.114 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.114 - 0.993i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (935, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.114 - 0.993i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2115418574 - 0.1884894713i\)
\(L(\frac12)\) \(\approx\) \(0.2115418574 - 0.1884894713i\)
\(L(1)\) \(\approx\) \(0.2786301036 - 0.3752125411i\)
\(L(1)\) \(\approx\) \(0.2786301036 - 0.3752125411i\)

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 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (-0.5 - 0.866i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + T \)
29 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + T \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 + T \)
59 \( 1 + (-0.5 + 0.866i)T \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 + (-0.5 - 0.866i)T \)
79 \( 1 + T \)
83 \( 1 + T \)
89 \( 1 + T \)
97 \( 1 + (-0.5 + 0.866i)T \)
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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

−18.47118276762174287596981625061, −17.79507613897115852934913983967, −17.26310726407070759632114992830, −16.545220138542765806309950229881, −15.656213593044527965537219198, −15.431371779488623730435091164685, −14.88752336912599971422518865588, −14.345323998236460139405901233487, −13.19331557124077998495625107962, −12.39650194020678124650180146139, −11.564454914887404137472429325392, −10.78549803612195097436379718163, −10.37949781198065268696173150457, −9.408438824303496782312103446737, −9.12440559919141052471416097453, −8.23746273619399637649242447699, −7.200085953146323828757166910523, −6.72439508222678083702997653208, −6.08871120237861393301847657355, −5.23257565711259468166623516384, −4.550201829237319654420353484510, −3.87098612130524198499122829651, −2.74251187789682321703153345129, −1.9484841161869483506530266070, −0.190458296227889946131154751753, 0.594598135952869542572229381341, 1.136951366100922616192431990789, 2.23373618129983143652611321762, 3.111517581716951564711520349129, 3.81927732893450556365752280999, 4.85741847714113513199588151066, 5.37599497213514924568159287773, 6.49561806706361488433282028349, 7.54438950234328475581307653907, 7.68392981892579341083121025150, 8.58604738008734585025922846489, 9.24643937288805574372756402554, 10.22773882526466493531954926488, 10.86074639224738398115888289909, 11.38215586202663778771413033833, 12.19436878003551636393974205795, 12.7985218646887985012375688533, 13.25486703944500080733459643275, 13.681027341887017942469423890237, 14.849868581315396763740378216389, 16.09526057498755515460478690182, 16.53060883506793037127418269006, 16.984665625152550040659975129935, 17.66224731466268285591804461345, 18.51119694689361276869074275754

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