Properties

Degree 2
Conductor $ 3^{2} \cdot 11 \cdot 61 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.76·2-s + 1.12·4-s + 2.50·5-s − 4.08·7-s + 1.55·8-s − 4.42·10-s + 11-s + 4.58·13-s + 7.22·14-s − 4.98·16-s + 1.68·17-s + 2.54·19-s + 2.80·20-s − 1.76·22-s − 6.64·23-s + 1.26·25-s − 8.09·26-s − 4.58·28-s − 2.87·29-s + 5.77·31-s + 5.69·32-s − 2.98·34-s − 10.2·35-s + 10.5·37-s − 4.49·38-s + 3.89·40-s + 7.50·41-s + ⋯
L(s)  = 1  − 1.24·2-s + 0.560·4-s + 1.11·5-s − 1.54·7-s + 0.549·8-s − 1.39·10-s + 0.301·11-s + 1.27·13-s + 1.93·14-s − 1.24·16-s + 0.409·17-s + 0.583·19-s + 0.626·20-s − 0.376·22-s − 1.38·23-s + 0.252·25-s − 1.58·26-s − 0.865·28-s − 0.533·29-s + 1.03·31-s + 1.00·32-s − 0.511·34-s − 1.72·35-s + 1.73·37-s − 0.728·38-s + 0.615·40-s + 1.17·41-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

\( d \)  =  \(2\)
\( N \)  =  \(6039\)    =    \(3^{2} \cdot 11 \cdot 61\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{6039} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 6039,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.035164585$
$L(\frac12)$  $\approx$  $1.035164585$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;11,\;61\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;11,\;61\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 \)
11 \( 1 - T \)
61 \( 1 - T \)
good2 \( 1 + 1.76T + 2T^{2} \)
5 \( 1 - 2.50T + 5T^{2} \)
7 \( 1 + 4.08T + 7T^{2} \)
13 \( 1 - 4.58T + 13T^{2} \)
17 \( 1 - 1.68T + 17T^{2} \)
19 \( 1 - 2.54T + 19T^{2} \)
23 \( 1 + 6.64T + 23T^{2} \)
29 \( 1 + 2.87T + 29T^{2} \)
31 \( 1 - 5.77T + 31T^{2} \)
37 \( 1 - 10.5T + 37T^{2} \)
41 \( 1 - 7.50T + 41T^{2} \)
43 \( 1 + 5.62T + 43T^{2} \)
47 \( 1 + 8.24T + 47T^{2} \)
53 \( 1 - 6.73T + 53T^{2} \)
59 \( 1 + 1.44T + 59T^{2} \)
67 \( 1 + 2.89T + 67T^{2} \)
71 \( 1 - 3.64T + 71T^{2} \)
73 \( 1 + 1.53T + 73T^{2} \)
79 \( 1 - 6.59T + 79T^{2} \)
83 \( 1 + 4.68T + 83T^{2} \)
89 \( 1 + 8.01T + 89T^{2} \)
97 \( 1 + 8.48T + 97T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.195918101616088783421692740494, −7.54870807292454596588395071308, −6.53532912928901095546535540561, −6.21042397070028902431211913954, −5.58570922285244273010918236277, −4.30720340134000214990759733727, −3.49922457885070717152230188482, −2.54003045720345250237665352345, −1.57545068657445943036227458046, −0.67828378037022359312773144964, 0.67828378037022359312773144964, 1.57545068657445943036227458046, 2.54003045720345250237665352345, 3.49922457885070717152230188482, 4.30720340134000214990759733727, 5.58570922285244273010918236277, 6.21042397070028902431211913954, 6.53532912928901095546535540561, 7.54870807292454596588395071308, 8.195918101616088783421692740494

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