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Properties

Label 1-4033-4033.3097-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: $\chi_{4033} (3097, \cdot )$
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

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

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