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-s − 4-s − 3.85·5-s − 7-s + 3·8-s + 3.85·10-s − 3.23·13-s + 14-s − 16-s + 1.14·17-s − 0.381·19-s + 3.85·20-s − 8.09·23-s + 9.85·25-s + 3.23·26-s + 28-s − 2·29-s + 8.32·31-s − 5·32-s − 1.14·34-s + 3.85·35-s − 0.909·37-s + 0.381·38-s − 11.5·40-s − 4.14·41-s − 3.23·43-s + 8.09·46-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.5·4-s − 1.72·5-s − 0.377·7-s + 1.06·8-s + 1.21·10-s − 0.897·13-s + 0.267·14-s − 0.250·16-s + 0.277·17-s − 0.0876·19-s + 0.861·20-s − 1.68·23-s + 1.97·25-s + 0.634·26-s + 0.188·28-s − 0.371·29-s + 1.49·31-s − 0.883·32-s − 0.196·34-s + 0.651·35-s − 0.149·37-s + 0.0619·38-s − 1.82·40-s − 0.647·41-s − 0.493·43-s + 1.19·46-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 \( 1 + T + 2T^{2} \)
5 \( 1 + 3.85T + 5T^{2} \)
13 \( 1 + 3.23T + 13T^{2} \)
17 \( 1 - 1.14T + 17T^{2} \)
19 \( 1 + 0.381T + 19T^{2} \)
23 \( 1 + 8.09T + 23T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 - 8.32T + 31T^{2} \)
37 \( 1 + 0.909T + 37T^{2} \)
41 \( 1 + 4.14T + 41T^{2} \)
43 \( 1 + 3.23T + 43T^{2} \)
47 \( 1 + 2T + 47T^{2} \)
53 \( 1 - 10.1T + 53T^{2} \)
59 \( 1 + 11.2T + 59T^{2} \)
61 \( 1 - 8.94T + 61T^{2} \)
67 \( 1 - 8T + 67T^{2} \)
71 \( 1 - 8.94T + 71T^{2} \)
73 \( 1 - 0.763T + 73T^{2} \)
79 \( 1 - 14T + 79T^{2} \)
83 \( 1 + 14.9T + 83T^{2} \)
89 \( 1 - 10.0T + 89T^{2} \)
97 \( 1 + 12.4T + 97T^{2} \)
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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

−7.66853476741148730248572221043, −7.18044414593632609202247797407, −6.35557063876303611309578568204, −5.22653768660608240930313322800, −4.55880766849690184720141595047, −3.94203535381861160856702839641, −3.32231877676726239253614514322, −2.15231024873850373622560961908, −0.77285977765209532936188379400, 0, 0.77285977765209532936188379400, 2.15231024873850373622560961908, 3.32231877676726239253614514322, 3.94203535381861160856702839641, 4.55880766849690184720141595047, 5.22653768660608240930313322800, 6.35557063876303611309578568204, 7.18044414593632609202247797407, 7.66853476741148730248572221043

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