Properties

Label 2-6039-1.1-c1-0-139
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  + 1.56·2-s + 0.452·4-s + 1.36·5-s + 3.35·7-s − 2.42·8-s + 2.13·10-s + 11-s + 6.49·13-s + 5.26·14-s − 4.70·16-s + 3.12·17-s − 0.766·19-s + 0.616·20-s + 1.56·22-s − 0.696·23-s − 3.14·25-s + 10.1·26-s + 1.52·28-s − 1.69·29-s − 0.344·31-s − 2.51·32-s + 4.90·34-s + 4.57·35-s + 5.23·37-s − 1.20·38-s − 3.29·40-s + 8.27·41-s + ⋯
L(s)  = 1  + 1.10·2-s + 0.226·4-s + 0.608·5-s + 1.26·7-s − 0.856·8-s + 0.674·10-s + 0.301·11-s + 1.80·13-s + 1.40·14-s − 1.17·16-s + 0.758·17-s − 0.175·19-s + 0.137·20-s + 0.333·22-s − 0.145·23-s − 0.629·25-s + 1.99·26-s + 0.287·28-s − 0.314·29-s − 0.0618·31-s − 0.444·32-s + 0.840·34-s + 0.773·35-s + 0.861·37-s − 0.194·38-s − 0.521·40-s + 1.29·41-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

\(L(1)\) \(\approx\) \(4.835124306\)
\(L(\frac12)\) \(\approx\) \(4.835124306\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.56T + 2T^{2} \)
5 \( 1 - 1.36T + 5T^{2} \)
7 \( 1 - 3.35T + 7T^{2} \)
13 \( 1 - 6.49T + 13T^{2} \)
17 \( 1 - 3.12T + 17T^{2} \)
19 \( 1 + 0.766T + 19T^{2} \)
23 \( 1 + 0.696T + 23T^{2} \)
29 \( 1 + 1.69T + 29T^{2} \)
31 \( 1 + 0.344T + 31T^{2} \)
37 \( 1 - 5.23T + 37T^{2} \)
41 \( 1 - 8.27T + 41T^{2} \)
43 \( 1 + 1.26T + 43T^{2} \)
47 \( 1 + 2.80T + 47T^{2} \)
53 \( 1 - 13.4T + 53T^{2} \)
59 \( 1 - 7.04T + 59T^{2} \)
67 \( 1 + 11.9T + 67T^{2} \)
71 \( 1 - 1.74T + 71T^{2} \)
73 \( 1 + 9.19T + 73T^{2} \)
79 \( 1 + 11.9T + 79T^{2} \)
83 \( 1 + 1.29T + 83T^{2} \)
89 \( 1 + 1.41T + 89T^{2} \)
97 \( 1 + 5.21T + 97T^{2} \)
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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

−8.139555765767110154226463697030, −7.28683160925386377453721592149, −6.21815842382916556907335963949, −5.82444956932895104275753170102, −5.29833372047371918228220003534, −4.29953660833978259539874106339, −3.94184677603905354436149346964, −2.97181322204935569934390064105, −1.92746984308308784130843525997, −1.07033867918641034614403619704, 1.07033867918641034614403619704, 1.92746984308308784130843525997, 2.97181322204935569934390064105, 3.94184677603905354436149346964, 4.29953660833978259539874106339, 5.29833372047371918228220003534, 5.82444956932895104275753170102, 6.21815842382916556907335963949, 7.28683160925386377453721592149, 8.139555765767110154226463697030

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