Properties

Degree 2
Conductor $ 2^{6} \cdot 3 \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  + 3-s + 2·7-s + 9-s − 4·13-s + 6·17-s + 4·19-s + 2·21-s − 6·23-s − 5·25-s + 27-s + 6·29-s − 8·31-s + 10·37-s − 4·39-s − 6·41-s − 8·43-s + 6·47-s − 3·49-s + 6·51-s + 4·57-s + 8·61-s + 2·63-s − 4·67-s − 6·69-s − 6·71-s − 2·73-s − 5·75-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.755·7-s + 1/3·9-s − 1.10·13-s + 1.45·17-s + 0.917·19-s + 0.436·21-s − 1.25·23-s − 25-s + 0.192·27-s + 1.11·29-s − 1.43·31-s + 1.64·37-s − 0.640·39-s − 0.937·41-s − 1.21·43-s + 0.875·47-s − 3/7·49-s + 0.840·51-s + 0.529·57-s + 1.02·61-s + 0.251·63-s − 0.488·67-s − 0.722·69-s − 0.712·71-s − 0.234·73-s − 0.577·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 23232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(23232\)    =    \(2^{6} \cdot 3 \cdot 11^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{23232} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 23232,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.058928454$
$L(\frac12)$  $\approx$  $3.058928454$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;11\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
11 \( 1 \)
good5 \( 1 + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−15.24346442917200, −14.89029409411051, −14.31402521107094, −14.04770668292028, −13.44833309689990, −12.73305681891186, −12.08123365304865, −11.81930481491102, −11.22897167832771, −10.24774654925016, −9.971314196720941, −9.535195649542273, −8.747813495878432, −8.072632677949418, −7.645650850364828, −7.370532458590688, −6.386112213477110, −5.675203003647681, −5.096544574754594, −4.511824091419983, −3.679187748937799, −3.147443397928570, −2.230318667073621, −1.690939782441019, −0.6812529208566823, 0.6812529208566823, 1.690939782441019, 2.230318667073621, 3.147443397928570, 3.679187748937799, 4.511824091419983, 5.096544574754594, 5.675203003647681, 6.386112213477110, 7.370532458590688, 7.645650850364828, 8.072632677949418, 8.747813495878432, 9.535195649542273, 9.971314196720941, 10.24774654925016, 11.22897167832771, 11.81930481491102, 12.08123365304865, 12.73305681891186, 13.44833309689990, 14.04770668292028, 14.31402521107094, 14.89029409411051, 15.24346442917200

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