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  =  $(2,\ 23232,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.333374487$
$L(\frac12)$  $\approx$  $1.333374487$
$L(\frac{3}{2})$   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

−15.58175686392737, −14.92972186164853, −14.42276780386654, −13.91678864289426, −12.96587652919719, −12.87954131236200, −12.21868801643870, −11.68039413661287, −11.24086698647830, −10.35085561673673, −9.895655538285737, −9.750389109866077, −8.841265258960583, −8.116837321751180, −7.610374457568187, −6.894011743562257, −6.399265769769514, −5.843717633559277, −5.113576960271904, −4.580740292573273, −3.859288921150765, −2.951559910398094, −2.519488677752431, −1.319619414161837, −0.5144853988055250, 0.5144853988055250, 1.319619414161837, 2.519488677752431, 2.951559910398094, 3.859288921150765, 4.580740292573273, 5.113576960271904, 5.843717633559277, 6.399265769769514, 6.894011743562257, 7.610374457568187, 8.116837321751180, 8.841265258960583, 9.750389109866077, 9.895655538285737, 10.35085561673673, 11.24086698647830, 11.68039413661287, 12.21868801643870, 12.87954131236200, 12.96587652919719, 13.91678864289426, 14.42276780386654, 14.92972186164853, 15.58175686392737

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