Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.12·2-s + 2.51·4-s − 0.484·5-s + 7-s + 1.09·8-s − 1.03·10-s + 5.60·13-s + 2.12·14-s − 2.70·16-s − 5.60·17-s − 5.28·19-s − 1.21·20-s − 2.48·23-s − 4.76·25-s + 11.9·26-s + 2.51·28-s − 5.28·29-s − 7.12·31-s − 7.93·32-s − 11.9·34-s − 0.484·35-s − 0.235·37-s − 11.2·38-s − 0.530·40-s − 2.39·41-s + 1.03·43-s − 5.28·46-s + ⋯
L(s)  = 1  + 1.50·2-s + 1.25·4-s − 0.216·5-s + 0.377·7-s + 0.387·8-s − 0.325·10-s + 1.55·13-s + 0.567·14-s − 0.676·16-s − 1.36·17-s − 1.21·19-s − 0.272·20-s − 0.518·23-s − 0.952·25-s + 2.33·26-s + 0.475·28-s − 0.980·29-s − 1.27·31-s − 1.40·32-s − 2.04·34-s − 0.0819·35-s − 0.0386·37-s − 1.82·38-s − 0.0839·40-s − 0.373·41-s + 0.157·43-s − 0.778·46-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  =  1
Selberg data  =  $(2,\ 7623,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 \( 1 - 2.12T + 2T^{2} \)
5 \( 1 + 0.484T + 5T^{2} \)
13 \( 1 - 5.60T + 13T^{2} \)
17 \( 1 + 5.60T + 17T^{2} \)
19 \( 1 + 5.28T + 19T^{2} \)
23 \( 1 + 2.48T + 23T^{2} \)
29 \( 1 + 5.28T + 29T^{2} \)
31 \( 1 + 7.12T + 31T^{2} \)
37 \( 1 + 0.235T + 37T^{2} \)
41 \( 1 + 2.39T + 41T^{2} \)
43 \( 1 - 1.03T + 43T^{2} \)
47 \( 1 - 1.60T + 47T^{2} \)
53 \( 1 - 3.03T + 53T^{2} \)
59 \( 1 + 3.12T + 59T^{2} \)
61 \( 1 + 2.39T + 61T^{2} \)
67 \( 1 - 10.0T + 67T^{2} \)
71 \( 1 + 12.0T + 71T^{2} \)
73 \( 1 - 2.39T + 73T^{2} \)
79 \( 1 - 9.03T + 79T^{2} \)
83 \( 1 - 3.21T + 83T^{2} \)
89 \( 1 - 1.26T + 89T^{2} \)
97 \( 1 + 8.79T + 97T^{2} \)
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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

−7.29609222558563396266763543826, −6.51237889931001551178237694924, −6.03648302897845718053503583793, −5.41422456700486232422723491694, −4.52718885811727722323314499442, −3.93486308076744415584564911596, −3.57297981054057704622203547003, −2.34208739529183519494339866583, −1.74635018172508328530954454478, 0, 1.74635018172508328530954454478, 2.34208739529183519494339866583, 3.57297981054057704622203547003, 3.93486308076744415584564911596, 4.52718885811727722323314499442, 5.41422456700486232422723491694, 6.03648302897845718053503583793, 6.51237889931001551178237694924, 7.29609222558563396266763543826

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