Properties

Degree 2
Conductor $ 3^{2} \cdot 7 \cdot 23^{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 − 2·5-s + 7-s − 3·8-s − 2·10-s + 4·11-s − 2·13-s + 14-s − 16-s − 6·17-s − 4·19-s + 2·20-s + 4·22-s − 25-s − 2·26-s − 28-s + 2·29-s + 5·32-s − 6·34-s − 2·35-s − 6·37-s − 4·38-s + 6·40-s − 2·41-s + 4·43-s − 4·44-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s − 0.894·5-s + 0.377·7-s − 1.06·8-s − 0.632·10-s + 1.20·11-s − 0.554·13-s + 0.267·14-s − 1/4·16-s − 1.45·17-s − 0.917·19-s + 0.447·20-s + 0.852·22-s − 1/5·25-s − 0.392·26-s − 0.188·28-s + 0.371·29-s + 0.883·32-s − 1.02·34-s − 0.338·35-s − 0.986·37-s − 0.648·38-s + 0.948·40-s − 0.312·41-s + 0.609·43-s − 0.603·44-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 33327 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 33327 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(33327\)    =    \(3^{2} \cdot 7 \cdot 23^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{33327} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 33327,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.8689266066$
$L(\frac12)$  $\approx$  $0.8689266066$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;7,\;23\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;7,\;23\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 \)
7 \( 1 - T \)
23 \( 1 \)
good2 \( 1 - T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−14.98965123574319, −14.50837438527650, −13.85271396623249, −13.63960656248035, −12.82498686361913, −12.35653456813616, −11.96435588166499, −11.47552855150514, −10.91932215965011, −10.31238839231699, −9.417369203897864, −9.119807297535601, −8.436082917442770, −8.163268786660262, −7.176799994749514, −6.777124866432225, −6.145251023360417, −5.452364010217351, −4.688348636630106, −4.221515233408947, −4.001117952713644, −3.157705580799969, −2.372253377790392, −1.487830945243176, −0.3147800536775853, 0.3147800536775853, 1.487830945243176, 2.372253377790392, 3.157705580799969, 4.001117952713644, 4.221515233408947, 4.688348636630106, 5.452364010217351, 6.145251023360417, 6.777124866432225, 7.176799994749514, 8.163268786660262, 8.436082917442770, 9.119807297535601, 9.417369203897864, 10.31238839231699, 10.91932215965011, 11.47552855150514, 11.96435588166499, 12.35653456813616, 12.82498686361913, 13.63960656248035, 13.85271396623249, 14.50837438527650, 14.98965123574319

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