Properties

Label 2-6039-1.1-c1-0-12
Degree $2$
Conductor $6039$
Sign $1$
Analytic cond. $48.2216$
Root an. cond. $6.94418$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.0892·2-s − 1.99·4-s + 0.250·5-s − 4.16·7-s − 0.356·8-s + 0.0223·10-s + 11-s − 2.99·13-s − 0.371·14-s + 3.95·16-s − 4.74·17-s − 0.0676·19-s − 0.499·20-s + 0.0892·22-s − 5.30·23-s − 4.93·25-s − 0.267·26-s + 8.29·28-s − 5.69·29-s − 1.66·31-s + 1.06·32-s − 0.423·34-s − 1.04·35-s − 3.66·37-s − 0.00604·38-s − 0.0893·40-s + 1.93·41-s + ⋯
L(s)  = 1  + 0.0631·2-s − 0.996·4-s + 0.112·5-s − 1.57·7-s − 0.125·8-s + 0.00707·10-s + 0.301·11-s − 0.830·13-s − 0.0993·14-s + 0.988·16-s − 1.15·17-s − 0.0155·19-s − 0.111·20-s + 0.0190·22-s − 1.10·23-s − 0.987·25-s − 0.0524·26-s + 1.56·28-s − 1.05·29-s − 0.299·31-s + 0.188·32-s − 0.0726·34-s − 0.176·35-s − 0.602·37-s − 0.000980·38-s − 0.0141·40-s + 0.301·41-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(6039\)    =    \(3^{2} \cdot 11 \cdot 61\)
Sign: $1$
Analytic conductor: \(48.2216\)
Root analytic conductor: \(6.94418\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6039,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3205024722\)
\(L(\frac12)\) \(\approx\) \(0.3205024722\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 - T \)
61 \( 1 + T \)
good2 \( 1 - 0.0892T + 2T^{2} \)
5 \( 1 - 0.250T + 5T^{2} \)
7 \( 1 + 4.16T + 7T^{2} \)
13 \( 1 + 2.99T + 13T^{2} \)
17 \( 1 + 4.74T + 17T^{2} \)
19 \( 1 + 0.0676T + 19T^{2} \)
23 \( 1 + 5.30T + 23T^{2} \)
29 \( 1 + 5.69T + 29T^{2} \)
31 \( 1 + 1.66T + 31T^{2} \)
37 \( 1 + 3.66T + 37T^{2} \)
41 \( 1 - 1.93T + 41T^{2} \)
43 \( 1 - 8.77T + 43T^{2} \)
47 \( 1 + 7.56T + 47T^{2} \)
53 \( 1 + 6.58T + 53T^{2} \)
59 \( 1 - 6.14T + 59T^{2} \)
67 \( 1 - 3.02T + 67T^{2} \)
71 \( 1 - 2.19T + 71T^{2} \)
73 \( 1 + 14.6T + 73T^{2} \)
79 \( 1 - 10.6T + 79T^{2} \)
83 \( 1 + 7.54T + 83T^{2} \)
89 \( 1 - 3.77T + 89T^{2} \)
97 \( 1 - 1.83T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.095417199565440412707212797316, −7.36648738141120931243222443813, −6.55472004727884722482330966650, −5.95592324316232371103709208697, −5.25872643778416749466523531420, −4.22947507782097385572067114494, −3.82103541785475869489018609091, −2.90663466929721343354412499167, −1.91143098370437170583172063386, −0.28024408363675081695801131889, 0.28024408363675081695801131889, 1.91143098370437170583172063386, 2.90663466929721343354412499167, 3.82103541785475869489018609091, 4.22947507782097385572067114494, 5.25872643778416749466523531420, 5.95592324316232371103709208697, 6.55472004727884722482330966650, 7.36648738141120931243222443813, 8.095417199565440412707212797316

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