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  − 0.226·2-s − 1.94·4-s + 2.49·5-s + 7-s + 0.893·8-s − 0.564·10-s + 5.13·13-s − 0.226·14-s + 3.69·16-s + 1.43·17-s − 6.06·19-s − 4.86·20-s − 7.08·23-s + 1.22·25-s − 1.16·26-s − 1.94·28-s − 6.51·29-s + 7.68·31-s − 2.62·32-s − 0.325·34-s + 2.49·35-s − 3.98·37-s + 1.37·38-s + 2.22·40-s − 6.74·41-s − 0.802·43-s + 1.60·46-s + ⋯
L(s)  = 1  − 0.159·2-s − 0.974·4-s + 1.11·5-s + 0.377·7-s + 0.315·8-s − 0.178·10-s + 1.42·13-s − 0.0604·14-s + 0.923·16-s + 0.348·17-s − 1.39·19-s − 1.08·20-s − 1.47·23-s + 0.245·25-s − 0.227·26-s − 0.368·28-s − 1.21·29-s + 1.38·31-s − 0.463·32-s − 0.0557·34-s + 0.421·35-s − 0.655·37-s + 0.222·38-s + 0.352·40-s − 1.05·41-s − 0.122·43-s + 0.236·46-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 \( 1 + 0.226T + 2T^{2} \)
5 \( 1 - 2.49T + 5T^{2} \)
13 \( 1 - 5.13T + 13T^{2} \)
17 \( 1 - 1.43T + 17T^{2} \)
19 \( 1 + 6.06T + 19T^{2} \)
23 \( 1 + 7.08T + 23T^{2} \)
29 \( 1 + 6.51T + 29T^{2} \)
31 \( 1 - 7.68T + 31T^{2} \)
37 \( 1 + 3.98T + 37T^{2} \)
41 \( 1 + 6.74T + 41T^{2} \)
43 \( 1 + 0.802T + 43T^{2} \)
47 \( 1 + 6.75T + 47T^{2} \)
53 \( 1 + 6.58T + 53T^{2} \)
59 \( 1 + 2.87T + 59T^{2} \)
61 \( 1 + 0.855T + 61T^{2} \)
67 \( 1 + 1.64T + 67T^{2} \)
71 \( 1 + 4.52T + 71T^{2} \)
73 \( 1 + 14.8T + 73T^{2} \)
79 \( 1 - 2.45T + 79T^{2} \)
83 \( 1 + 2.24T + 83T^{2} \)
89 \( 1 + 1.73T + 89T^{2} \)
97 \( 1 + 12.0T + 97T^{2} \)
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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

−7.81247904037742838629097814505, −6.63166241310947704267442013285, −6.00548136872422538086545953522, −5.58468019148770066352865387780, −4.64714896590643359512915173481, −4.03734172481134501271514780483, −3.22225066375480446145569439808, −1.92192425088444527438095896253, −1.43039474556594633313353251883, 0, 1.43039474556594633313353251883, 1.92192425088444527438095896253, 3.22225066375480446145569439808, 4.03734172481134501271514780483, 4.64714896590643359512915173481, 5.58468019148770066352865387780, 6.00548136872422538086545953522, 6.63166241310947704267442013285, 7.81247904037742838629097814505

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