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-s − 4-s − 5-s + 7-s − 3·8-s − 10-s + 5·13-s + 14-s − 16-s + 7·17-s − 6·19-s + 20-s + 4·23-s − 4·25-s + 5·26-s − 28-s − 9·29-s − 2·31-s + 5·32-s + 7·34-s − 35-s + 9·37-s − 6·38-s + 3·40-s + 7·41-s − 6·43-s + 4·46-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s − 0.447·5-s + 0.377·7-s − 1.06·8-s − 0.316·10-s + 1.38·13-s + 0.267·14-s − 1/4·16-s + 1.69·17-s − 1.37·19-s + 0.223·20-s + 0.834·23-s − 4/5·25-s + 0.980·26-s − 0.188·28-s − 1.67·29-s − 0.359·31-s + 0.883·32-s + 1.20·34-s − 0.169·35-s + 1.47·37-s − 0.973·38-s + 0.474·40-s + 1.09·41-s − 0.914·43-s + 0.589·46-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  :  $\chi_{7623} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 7623,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.195090343$
$L(\frac12)$  $\approx$  $2.195090343$
$L(\frac{3}{2})$   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 \( 1 - T + p T^{2} \)
5 \( 1 + T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 - 7 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 9 T + p T^{2} \)
41 \( 1 - 7 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 + 2 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 9 T + p T^{2} \)
97 \( 1 - 17 T + p T^{2} \)
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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

−17.01241056148295, −16.45625936976373, −15.78022141371471, −15.05818638349663, −14.65611607544575, −14.27446820785078, −13.36963743647035, −12.96643608024743, −12.59974602537041, −11.51929662810889, −11.43973517425438, −10.55497885351298, −9.783930992160781, −9.105929343622684, −8.483637320866558, −7.924542437429133, −7.254614091710017, −6.009513018960009, −5.927676082703965, −4.995158907853685, −4.199046758444023, −3.707667361648841, −3.059481994057702, −1.786392294454110, −0.6830346549159239, 0.6830346549159239, 1.786392294454110, 3.059481994057702, 3.707667361648841, 4.199046758444023, 4.995158907853685, 5.927676082703965, 6.009513018960009, 7.254614091710017, 7.924542437429133, 8.483637320866558, 9.105929343622684, 9.783930992160781, 10.55497885351298, 11.43973517425438, 11.51929662810889, 12.59974602537041, 12.96643608024743, 13.36963743647035, 14.27446820785078, 14.65611607544575, 15.05818638349663, 15.78022141371471, 16.45625936976373, 17.01241056148295

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