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Properties

Label 1-6039-6039.43-r0-0-0
Degree $1$
Conductor $6039$
Sign $0.180 + 0.983i$
Analytic cond. $28.0449$
Root an. cond. $28.0449$
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  + (0.743 − 0.669i)2-s + (0.104 − 0.994i)4-s + (0.809 + 0.587i)5-s + (−0.207 + 0.978i)7-s + (−0.587 − 0.809i)8-s + (0.994 − 0.104i)10-s − 13-s + (0.5 + 0.866i)14-s + (−0.978 − 0.207i)16-s + (0.406 + 0.913i)17-s + (0.669 + 0.743i)19-s + (0.669 − 0.743i)20-s + (0.406 + 0.913i)23-s + (0.309 + 0.951i)25-s + (−0.743 + 0.669i)26-s + ⋯
L(s)  = 1  + (0.743 − 0.669i)2-s + (0.104 − 0.994i)4-s + (0.809 + 0.587i)5-s + (−0.207 + 0.978i)7-s + (−0.587 − 0.809i)8-s + (0.994 − 0.104i)10-s − 13-s + (0.5 + 0.866i)14-s + (−0.978 − 0.207i)16-s + (0.406 + 0.913i)17-s + (0.669 + 0.743i)19-s + (0.669 − 0.743i)20-s + (0.406 + 0.913i)23-s + (0.309 + 0.951i)25-s + (−0.743 + 0.669i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.180 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.180 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6039\)    =    \(3^{2} \cdot 11 \cdot 61\)
Sign: $0.180 + 0.983i$
Analytic conductor: \(28.0449\)
Root analytic conductor: \(28.0449\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6039} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6039,\ (0:\ ),\ 0.180 + 0.983i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.601601749 + 1.334962759i\)
\(L(\frac12)\) \(\approx\) \(1.601601749 + 1.334962759i\)
\(L(1)\) \(\approx\) \(1.524112215 - 0.09299955823i\)
\(L(1)\) \(\approx\) \(1.524112215 - 0.09299955823i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
61 \( 1 \)
good2 \( 1 + (0.743 - 0.669i)T \)
5 \( 1 + (0.809 + 0.587i)T \)
7 \( 1 + (-0.207 + 0.978i)T \)
13 \( 1 - T \)
17 \( 1 + (0.406 + 0.913i)T \)
19 \( 1 + (0.669 + 0.743i)T \)
23 \( 1 + (0.406 + 0.913i)T \)
29 \( 1 + iT \)
31 \( 1 + (-0.743 - 0.669i)T \)
37 \( 1 + (0.951 - 0.309i)T \)
41 \( 1 + (-0.978 - 0.207i)T \)
43 \( 1 + (0.994 - 0.104i)T \)
47 \( 1 + T \)
53 \( 1 + (-0.587 - 0.809i)T \)
59 \( 1 + (-0.951 - 0.309i)T \)
67 \( 1 + (-0.587 + 0.809i)T \)
71 \( 1 + (-0.994 + 0.104i)T \)
73 \( 1 + (0.104 + 0.994i)T \)
79 \( 1 + (-0.994 + 0.104i)T \)
83 \( 1 + (-0.669 - 0.743i)T \)
89 \( 1 + (-0.951 - 0.309i)T \)
97 \( 1 + (-0.309 - 0.951i)T \)
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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

−17.324468943958976986237242821642, −16.75734359950453664980833594594, −16.42373982081554763316742105739, −15.630241429950099204768285521281, −14.841069394709600454601981980488, −14.06426765187613957951387349487, −13.79513587037670040450512811921, −13.114500013552589771866003295636, −12.48624925690750185381342458486, −11.90544387905260726807960899885, −11.01153792732519746372238706089, −10.198217004930723605811462695048, −9.42159727228170852938949693467, −8.984230197672433186687550834692, −7.8877512263988063849304936767, −7.38816248541869424510525134539, −6.75601060786940469713910356556, −6.02549800465827100317398617237, −5.22809595333120169392361408553, −4.68580206494384439586029344312, −4.172167747417949539410161310252, −2.95316529288592971409041793122, −2.62417565622541350232373944618, −1.38879848796836049883491884435, −0.35570555982552486466746141155, 1.36225611055797246117982248780, 1.9012376113936624973256976536, 2.71312334212530836115420282519, 3.2064325427755794577327773711, 4.02778918610948072385244077272, 5.1002736214203669198222270937, 5.664090132192475911643361661468, 5.97019509311980320194883115942, 6.95231255100896033001762909699, 7.59847061560911657000762630550, 8.821960257448541713887731700668, 9.46189621990255628951443025602, 9.9484325707955576136225038205, 10.57370284412456247512087466977, 11.37831107443139855176726190894, 11.94072859963816001614912154656, 12.80601552595783594780846652709, 12.946537287716748514189192052919, 14.083618533648818078049680121802, 14.43942562915874603186067717961, 15.02223742483850718236186063707, 15.595680615334616185698447246188, 16.51150868882387501138373554182, 17.30057968214133839575379737716, 18.03071888359886849781156996084

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