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 − 2·5-s − 2·7-s + 9-s − 2·13-s − 2·15-s − 4·17-s + 6·19-s − 2·21-s − 25-s + 27-s − 8·29-s + 8·31-s + 4·35-s − 10·37-s − 2·39-s − 8·41-s + 2·43-s − 2·45-s + 8·47-s − 3·49-s − 4·51-s + 2·53-s + 6·57-s + 12·59-s + 10·61-s − 2·63-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.894·5-s − 0.755·7-s + 1/3·9-s − 0.554·13-s − 0.516·15-s − 0.970·17-s + 1.37·19-s − 0.436·21-s − 1/5·25-s + 0.192·27-s − 1.48·29-s + 1.43·31-s + 0.676·35-s − 1.64·37-s − 0.320·39-s − 1.24·41-s + 0.304·43-s − 0.298·45-s + 1.16·47-s − 3/7·49-s − 0.560·51-s + 0.274·53-s + 0.794·57-s + 1.56·59-s + 1.28·61-s − 0.251·63-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  =  $(2,\ 23232,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.132662558$
$L(\frac12)$  $\approx$  $1.132662558$
$L(\frac{3}{2})$   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 \( 1 + 2 T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−15.53647756549816, −15.08089940195046, −14.37364534566734, −13.80801102438451, −13.36462890455934, −12.80449700272158, −12.19727851175077, −11.64646583605355, −11.30866572440188, −10.38602471469692, −9.873785796762350, −9.465410351283960, −8.691566421258473, −8.314627767595118, −7.579953137280886, −6.987206803442936, −6.775481900021464, −5.632293985245927, −5.177478266450105, −4.184049275197444, −3.835269266918468, −3.097832444102060, −2.513987030645233, −1.570816098943795, −0.4061485270314329, 0.4061485270314329, 1.570816098943795, 2.513987030645233, 3.097832444102060, 3.835269266918468, 4.184049275197444, 5.177478266450105, 5.632293985245927, 6.775481900021464, 6.987206803442936, 7.579953137280886, 8.314627767595118, 8.691566421258473, 9.465410351283960, 9.873785796762350, 10.38602471469692, 11.30866572440188, 11.64646583605355, 12.19727851175077, 12.80449700272158, 13.36462890455934, 13.80801102438451, 14.37364534566734, 15.08089940195046, 15.53647756549816

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