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Properties

Label 1-6039-6039.149-r0-0-0
Degree $1$
Conductor $6039$
Sign $0.840 - 0.541i$
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  + (0.978 + 0.207i)2-s + (0.913 + 0.406i)4-s + (−0.669 + 0.743i)5-s + (−0.5 + 0.866i)7-s + (0.809 + 0.587i)8-s + (−0.809 + 0.587i)10-s + (−0.913 − 0.406i)13-s + (−0.669 + 0.743i)14-s + (0.669 + 0.743i)16-s + (−0.309 − 0.951i)17-s − 19-s + (−0.913 + 0.406i)20-s + (−0.104 + 0.994i)23-s + (−0.104 − 0.994i)25-s + (−0.809 − 0.587i)26-s + ⋯
L(s)  = 1  + (0.978 + 0.207i)2-s + (0.913 + 0.406i)4-s + (−0.669 + 0.743i)5-s + (−0.5 + 0.866i)7-s + (0.809 + 0.587i)8-s + (−0.809 + 0.587i)10-s + (−0.913 − 0.406i)13-s + (−0.669 + 0.743i)14-s + (0.669 + 0.743i)16-s + (−0.309 − 0.951i)17-s − 19-s + (−0.913 + 0.406i)20-s + (−0.104 + 0.994i)23-s + (−0.104 − 0.994i)25-s + (−0.809 − 0.587i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.840 - 0.541i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.840 - 0.541i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6039\)    =    \(3^{2} \cdot 11 \cdot 61\)
Sign: $0.840 - 0.541i$
Analytic conductor: \(28.0449\)
Root analytic conductor: \(28.0449\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6039} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6039,\ (0:\ ),\ 0.840 - 0.541i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.433335324 - 0.4215214813i\)
\(L(\frac12)\) \(\approx\) \(1.433335324 - 0.4215214813i\)
\(L(1)\) \(\approx\) \(1.310855320 + 0.3609559527i\)
\(L(1)\) \(\approx\) \(1.310855320 + 0.3609559527i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
61 \( 1 \)
good2 \( 1 + (0.978 + 0.207i)T \)
5 \( 1 + (-0.669 + 0.743i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (-0.913 - 0.406i)T \)
17 \( 1 + (-0.309 - 0.951i)T \)
19 \( 1 - T \)
23 \( 1 + (-0.104 + 0.994i)T \)
29 \( 1 + (-0.669 - 0.743i)T \)
31 \( 1 + (0.104 - 0.994i)T \)
37 \( 1 - T \)
41 \( 1 + (0.913 + 0.406i)T \)
43 \( 1 + (-0.104 - 0.994i)T \)
47 \( 1 + (0.978 + 0.207i)T \)
53 \( 1 + (0.309 - 0.951i)T \)
59 \( 1 + (-0.104 + 0.994i)T \)
67 \( 1 + (0.104 - 0.994i)T \)
71 \( 1 + T \)
73 \( 1 + (-0.309 - 0.951i)T \)
79 \( 1 + (0.669 + 0.743i)T \)
83 \( 1 + (-0.104 - 0.994i)T \)
89 \( 1 + (0.309 - 0.951i)T \)
97 \( 1 + (0.913 - 0.406i)T \)
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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

−17.401475415205511163632045736713, −16.94362927314427223520868197283, −16.358915105202368125284869590542, −15.80733228670335138939944862777, −15.02518030938070255048895741366, −14.47476243826274526859906789659, −13.81396313345710670216460063442, −12.96542921382999063498842730502, −12.51918357031532658266364029831, −12.224762593678428805293811727888, −11.11121071110161602799840935521, −10.71338934794485095076691634085, −10.021066123866456796130132312953, −9.111321917054370646641456559006, −8.34735323683803885043723431991, −7.49028153988671058831037587623, −6.90373292988405436026749674642, −6.30924943497948223712566601249, −5.343908200216829330701628174253, −4.62625225931926346724888471828, −4.104706874771226506464490299668, −3.60187514581896867580443178488, −2.61379810306351446177205480562, −1.772255791983928788635020413952, −0.88962805820286125301321084942, 0.27980521534347570206292280741, 2.126670272680808679169445927407, 2.42963019452903432943674001095, 3.24946220235975065263310777289, 3.87995534217930783167431601212, 4.67697985906504404381990886262, 5.46777788787830987420672405568, 6.08336488269089069112638897451, 6.80963379096276896586475009194, 7.44746507203897279043839034998, 7.95939233768390119866181131865, 8.93993706764996678882643111178, 9.7539933579292571367402532913, 10.55440792954193611621218670957, 11.27170925334953601291794518167, 11.90565481044252098032406797935, 12.26841630938563825400238661088, 13.11480907663739609299358602458, 13.67959187795862572733698158962, 14.55467776576200125537707012476, 15.07573935600014941430053326803, 15.50551744023168422140047149630, 15.98842704555843651996638934908, 16.89134052980644581357891064132, 17.49875941857631426920292951884

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