Properties

Label 2-6039-1.1-c1-0-121
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  − 1.53·2-s + 0.360·4-s + 2.16·5-s + 2.96·7-s + 2.51·8-s − 3.32·10-s − 11-s + 1.11·13-s − 4.55·14-s − 4.59·16-s + 6.92·17-s + 6.93·19-s + 0.780·20-s + 1.53·22-s − 0.602·23-s − 0.303·25-s − 1.71·26-s + 1.06·28-s + 4.99·29-s − 1.25·31-s + 2.01·32-s − 10.6·34-s + 6.42·35-s − 3.94·37-s − 10.6·38-s + 5.45·40-s − 1.36·41-s + ⋯
L(s)  = 1  − 1.08·2-s + 0.180·4-s + 0.969·5-s + 1.12·7-s + 0.890·8-s − 1.05·10-s − 0.301·11-s + 0.310·13-s − 1.21·14-s − 1.14·16-s + 1.67·17-s + 1.59·19-s + 0.174·20-s + 0.327·22-s − 0.125·23-s − 0.0606·25-s − 0.336·26-s + 0.201·28-s + 0.926·29-s − 0.224·31-s + 0.356·32-s − 1.82·34-s + 1.08·35-s − 0.648·37-s − 1.72·38-s + 0.863·40-s − 0.213·41-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

\(L(1)\) \(\approx\) \(1.773471706\)
\(L(\frac12)\) \(\approx\) \(1.773471706\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.53T + 2T^{2} \)
5 \( 1 - 2.16T + 5T^{2} \)
7 \( 1 - 2.96T + 7T^{2} \)
13 \( 1 - 1.11T + 13T^{2} \)
17 \( 1 - 6.92T + 17T^{2} \)
19 \( 1 - 6.93T + 19T^{2} \)
23 \( 1 + 0.602T + 23T^{2} \)
29 \( 1 - 4.99T + 29T^{2} \)
31 \( 1 + 1.25T + 31T^{2} \)
37 \( 1 + 3.94T + 37T^{2} \)
41 \( 1 + 1.36T + 41T^{2} \)
43 \( 1 - 11.4T + 43T^{2} \)
47 \( 1 + 3.48T + 47T^{2} \)
53 \( 1 - 9.97T + 53T^{2} \)
59 \( 1 - 4.08T + 59T^{2} \)
67 \( 1 + 5.56T + 67T^{2} \)
71 \( 1 - 6.01T + 71T^{2} \)
73 \( 1 - 4.24T + 73T^{2} \)
79 \( 1 + 15.1T + 79T^{2} \)
83 \( 1 - 10.9T + 83T^{2} \)
89 \( 1 + 3.58T + 89T^{2} \)
97 \( 1 + 15.8T + 97T^{2} \)
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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

−8.100075218223681499787601525343, −7.62707581450196012430360743108, −6.97949027759392461658968668995, −5.73367117352204855746245170128, −5.40094389584159608331920895977, −4.61702907170420466918680099257, −3.54721021229271543130194616187, −2.44286785221980939895758803172, −1.47407885315885811244071152471, −0.960749943208063925370280448197, 0.960749943208063925370280448197, 1.47407885315885811244071152471, 2.44286785221980939895758803172, 3.54721021229271543130194616187, 4.61702907170420466918680099257, 5.40094389584159608331920895977, 5.73367117352204855746245170128, 6.97949027759392461658968668995, 7.62707581450196012430360743108, 8.100075218223681499787601525343

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