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  − 1.69·2-s + 0.872·4-s − 1.13·5-s + 4.14·7-s + 1.91·8-s + 1.92·10-s + 11-s + 2.80·13-s − 7.02·14-s − 4.98·16-s + 0.763·17-s + 3.34·19-s − 0.991·20-s − 1.69·22-s − 3.86·23-s − 3.70·25-s − 4.76·26-s + 3.61·28-s − 2.17·29-s + 6.15·31-s + 4.62·32-s − 1.29·34-s − 4.71·35-s + 3.38·37-s − 5.67·38-s − 2.17·40-s − 1.67·41-s + ⋯
L(s)  = 1  − 1.19·2-s + 0.436·4-s − 0.508·5-s + 1.56·7-s + 0.675·8-s + 0.609·10-s + 0.301·11-s + 0.778·13-s − 1.87·14-s − 1.24·16-s + 0.185·17-s + 0.768·19-s − 0.221·20-s − 0.361·22-s − 0.805·23-s − 0.741·25-s − 0.933·26-s + 0.683·28-s − 0.403·29-s + 1.10·31-s + 0.817·32-s − 0.222·34-s − 0.796·35-s + 0.556·37-s − 0.920·38-s − 0.343·40-s − 0.261·41-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  =  $(2,\ 6039,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.236077529$
$L(\frac12)$  $\approx$  $1.236077529$
$L(\frac{3}{2})$   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 \( 1 + 1.69T + 2T^{2} \)
5 \( 1 + 1.13T + 5T^{2} \)
7 \( 1 - 4.14T + 7T^{2} \)
13 \( 1 - 2.80T + 13T^{2} \)
17 \( 1 - 0.763T + 17T^{2} \)
19 \( 1 - 3.34T + 19T^{2} \)
23 \( 1 + 3.86T + 23T^{2} \)
29 \( 1 + 2.17T + 29T^{2} \)
31 \( 1 - 6.15T + 31T^{2} \)
37 \( 1 - 3.38T + 37T^{2} \)
41 \( 1 + 1.67T + 41T^{2} \)
43 \( 1 - 5.23T + 43T^{2} \)
47 \( 1 - 12.4T + 47T^{2} \)
53 \( 1 + 4.85T + 53T^{2} \)
59 \( 1 - 11.2T + 59T^{2} \)
67 \( 1 - 3.11T + 67T^{2} \)
71 \( 1 - 0.313T + 71T^{2} \)
73 \( 1 - 4.11T + 73T^{2} \)
79 \( 1 - 0.122T + 79T^{2} \)
83 \( 1 - 2.98T + 83T^{2} \)
89 \( 1 + 12.3T + 89T^{2} \)
97 \( 1 - 0.287T + 97T^{2} \)
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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

−8.157007345263597107699264861806, −7.68608164448940533165979931500, −7.06715581000059049153318378298, −5.98793947150436004674180280316, −5.20742363949345805520265302069, −4.31247008763075131206819103304, −3.85073047683291332746401194588, −2.40447883003168788362274950876, −1.49517454072277944232370809450, −0.78339505709521181828766037710, 0.78339505709521181828766037710, 1.49517454072277944232370809450, 2.40447883003168788362274950876, 3.85073047683291332746401194588, 4.31247008763075131206819103304, 5.20742363949345805520265302069, 5.98793947150436004674180280316, 7.06715581000059049153318378298, 7.68608164448940533165979931500, 8.157007345263597107699264861806

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