Properties

Degree 2
Conductor $ 3^{2} \cdot 7 \cdot 11^{2} $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.39·2-s + 3.73·4-s + 3.93·5-s + 7-s − 4.14·8-s − 9.42·10-s + 2.99·13-s − 2.39·14-s + 2.45·16-s − 6.60·17-s + 5.90·19-s + 14.6·20-s + 6.02·23-s + 10.5·25-s − 7.17·26-s + 3.73·28-s − 1.52·29-s + 8.46·31-s + 2.40·32-s + 15.8·34-s + 3.93·35-s − 0.607·37-s − 14.1·38-s − 16.3·40-s − 1.70·41-s + 3.23·43-s − 14.4·46-s + ⋯
L(s)  = 1  − 1.69·2-s + 1.86·4-s + 1.76·5-s + 0.377·7-s − 1.46·8-s − 2.98·10-s + 0.831·13-s − 0.639·14-s + 0.614·16-s − 1.60·17-s + 1.35·19-s + 3.28·20-s + 1.25·23-s + 2.10·25-s − 1.40·26-s + 0.705·28-s − 0.282·29-s + 1.52·31-s + 0.424·32-s + 2.71·34-s + 0.665·35-s − 0.0999·37-s − 2.29·38-s − 2.58·40-s − 0.266·41-s + 0.493·43-s − 2.12·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(7623\)    =    \(3^{2} \cdot 7 \cdot 11^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{7623} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 7623,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.602228686$
$L(\frac12)$  $\approx$  $1.602228686$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;7,\;11\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 - T \)
11 \( 1 \)
good2 \( 1 + 2.39T + 2T^{2} \)
5 \( 1 - 3.93T + 5T^{2} \)
13 \( 1 - 2.99T + 13T^{2} \)
17 \( 1 + 6.60T + 17T^{2} \)
19 \( 1 - 5.90T + 19T^{2} \)
23 \( 1 - 6.02T + 23T^{2} \)
29 \( 1 + 1.52T + 29T^{2} \)
31 \( 1 - 8.46T + 31T^{2} \)
37 \( 1 + 0.607T + 37T^{2} \)
41 \( 1 + 1.70T + 41T^{2} \)
43 \( 1 - 3.23T + 43T^{2} \)
47 \( 1 - 4.80T + 47T^{2} \)
53 \( 1 - 6.12T + 53T^{2} \)
59 \( 1 + 6.23T + 59T^{2} \)
61 \( 1 - 2.08T + 61T^{2} \)
67 \( 1 + 0.599T + 67T^{2} \)
71 \( 1 + 1.40T + 71T^{2} \)
73 \( 1 + 7.08T + 73T^{2} \)
79 \( 1 - 1.02T + 79T^{2} \)
83 \( 1 - 3.08T + 83T^{2} \)
89 \( 1 - 2.48T + 89T^{2} \)
97 \( 1 - 2.55T + 97T^{2} \)
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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

−8.129844760710982314508477911577, −7.16024661451463712324281853746, −6.71209171606926494206670693702, −6.04227105964025401467454392195, −5.32810343983897761353802382063, −4.45884951003760506790717199610, −2.96424391970384107784996441060, −2.33952851171055968153378125752, −1.50661950918317011526586963203, −0.904534672416759829732917227783, 0.904534672416759829732917227783, 1.50661950918317011526586963203, 2.33952851171055968153378125752, 2.96424391970384107784996441060, 4.45884951003760506790717199610, 5.32810343983897761353802382063, 6.04227105964025401467454392195, 6.71209171606926494206670693702, 7.16024661451463712324281853746, 8.129844760710982314508477911577

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