Properties

Degree 2
Conductor $ 2^{4} \cdot 3 \cdot 11 \cdot 13^{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 + 4·7-s + 9-s + 11-s + 2·15-s − 2·17-s + 4·21-s − 8·23-s − 25-s + 27-s − 6·29-s − 8·31-s + 33-s + 8·35-s − 6·37-s + 2·41-s + 2·45-s + 8·47-s + 9·49-s − 2·51-s + 6·53-s + 2·55-s − 4·59-s + 6·61-s + 4·63-s − 4·67-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.894·5-s + 1.51·7-s + 1/3·9-s + 0.301·11-s + 0.516·15-s − 0.485·17-s + 0.872·21-s − 1.66·23-s − 1/5·25-s + 0.192·27-s − 1.11·29-s − 1.43·31-s + 0.174·33-s + 1.35·35-s − 0.986·37-s + 0.312·41-s + 0.298·45-s + 1.16·47-s + 9/7·49-s − 0.280·51-s + 0.824·53-s + 0.269·55-s − 0.520·59-s + 0.768·61-s + 0.503·63-s − 0.488·67-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 89232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 89232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(89232\)    =    \(2^{4} \cdot 3 \cdot 11 \cdot 13^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{89232} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 89232,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $4.567345985$
$L(\frac12)$  $\approx$  $4.567345985$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

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

−14.00227178585099, −13.49430499822353, −13.09998233503447, −12.29212243826144, −12.00100852875303, −11.37211272266638, −10.77296274337130, −10.54813388013989, −9.763391364805125, −9.356114023755706, −8.892603425617252, −8.372755130988536, −7.815144950135999, −7.454489783626837, −6.805682021027436, −6.114363555716086, −5.502599230550389, −5.255753303061556, −4.356620768322269, −3.992567322325012, −3.375800374011306, −2.286692260382097, −1.942304620143648, −1.703317557235185, −0.6224327537246720, 0.6224327537246720, 1.703317557235185, 1.942304620143648, 2.286692260382097, 3.375800374011306, 3.992567322325012, 4.356620768322269, 5.255753303061556, 5.502599230550389, 6.114363555716086, 6.805682021027436, 7.454489783626837, 7.815144950135999, 8.372755130988536, 8.892603425617252, 9.356114023755706, 9.763391364805125, 10.54813388013989, 10.77296274337130, 11.37211272266638, 12.00100852875303, 12.29212243826144, 13.09998233503447, 13.49430499822353, 14.00227178585099

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