Properties

Label 2-6039-1.1-c1-0-168
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.84·2-s + 1.41·4-s + 3.30·5-s − 3.95·7-s + 1.07·8-s − 6.10·10-s + 11-s − 2.98·13-s + 7.31·14-s − 4.82·16-s + 2.21·17-s + 3.56·19-s + 4.68·20-s − 1.84·22-s + 4.72·23-s + 5.91·25-s + 5.52·26-s − 5.60·28-s − 9.75·29-s − 4.49·31-s + 6.76·32-s − 4.09·34-s − 13.0·35-s − 5.68·37-s − 6.58·38-s + 3.56·40-s + 1.33·41-s + ⋯
L(s)  = 1  − 1.30·2-s + 0.708·4-s + 1.47·5-s − 1.49·7-s + 0.381·8-s − 1.93·10-s + 0.301·11-s − 0.829·13-s + 1.95·14-s − 1.20·16-s + 0.536·17-s + 0.817·19-s + 1.04·20-s − 0.394·22-s + 0.985·23-s + 1.18·25-s + 1.08·26-s − 1.05·28-s − 1.81·29-s − 0.806·31-s + 1.19·32-s − 0.701·34-s − 2.20·35-s − 0.934·37-s − 1.06·38-s + 0.563·40-s + 0.209·41-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: \(1\)
Selberg data: \((2,\ 6039,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.84T + 2T^{2} \)
5 \( 1 - 3.30T + 5T^{2} \)
7 \( 1 + 3.95T + 7T^{2} \)
13 \( 1 + 2.98T + 13T^{2} \)
17 \( 1 - 2.21T + 17T^{2} \)
19 \( 1 - 3.56T + 19T^{2} \)
23 \( 1 - 4.72T + 23T^{2} \)
29 \( 1 + 9.75T + 29T^{2} \)
31 \( 1 + 4.49T + 31T^{2} \)
37 \( 1 + 5.68T + 37T^{2} \)
41 \( 1 - 1.33T + 41T^{2} \)
43 \( 1 + 1.84T + 43T^{2} \)
47 \( 1 - 3.92T + 47T^{2} \)
53 \( 1 - 8.83T + 53T^{2} \)
59 \( 1 + 14.2T + 59T^{2} \)
67 \( 1 - 11.8T + 67T^{2} \)
71 \( 1 - 10.1T + 71T^{2} \)
73 \( 1 + 7.19T + 73T^{2} \)
79 \( 1 - 7.49T + 79T^{2} \)
83 \( 1 + 15.2T + 83T^{2} \)
89 \( 1 - 4.27T + 89T^{2} \)
97 \( 1 + 8.72T + 97T^{2} \)
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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

−7.64869735986997745031487251490, −7.08783456089141026010253048742, −6.57034585842347063945155648148, −5.62566001608262010689280787212, −5.20165282184055975287331195300, −3.83686265969941626270359954957, −2.92450979125443948647252893376, −2.08298901736170858778461382856, −1.19845981019679618926427575386, 0, 1.19845981019679618926427575386, 2.08298901736170858778461382856, 2.92450979125443948647252893376, 3.83686265969941626270359954957, 5.20165282184055975287331195300, 5.62566001608262010689280787212, 6.57034585842347063945155648148, 7.08783456089141026010253048742, 7.64869735986997745031487251490

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