Properties

Degree 2
Conductor $ 2^{6} \cdot 3 \cdot 7^{2} \cdot 11 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 9-s − 11-s − 4·13-s + 6·17-s + 4·19-s − 6·23-s − 5·25-s − 27-s − 6·29-s + 8·31-s + 33-s + 10·37-s + 4·39-s − 6·41-s + 8·43-s − 6·47-s − 6·51-s − 4·57-s + 8·61-s − 4·67-s + 6·69-s − 6·71-s − 2·73-s + 5·75-s − 14·79-s + 81-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s − 0.301·11-s − 1.10·13-s + 1.45·17-s + 0.917·19-s − 1.25·23-s − 25-s − 0.192·27-s − 1.11·29-s + 1.43·31-s + 0.174·33-s + 1.64·37-s + 0.640·39-s − 0.937·41-s + 1.21·43-s − 0.875·47-s − 0.840·51-s − 0.529·57-s + 1.02·61-s − 0.488·67-s + 0.722·69-s − 0.712·71-s − 0.234·73-s + 0.577·75-s − 1.57·79-s + 1/9·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 103488 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 103488 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(103488\)    =    \(2^{6} \cdot 3 \cdot 7^{2} \cdot 11\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{103488} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 103488,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7,\;11\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
7 \( 1 \)
11 \( 1 + T \)
good5 \( 1 + p T^{2} \)
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

−14.06260866986688, −13.38097358750198, −13.00424950468638, −12.34758421736065, −11.91259529806384, −11.71185026354661, −11.16275143890295, −10.37478607598978, −9.924872460723710, −9.784194546825599, −9.244011167752890, −8.306198296200274, −7.765729106305633, −7.639476476463520, −6.989514676440571, −6.244883137809137, −5.732342282658853, −5.457485798712125, −4.747399800313872, −4.238943282401341, −3.586339802171489, −2.914690135474110, −2.297956737697096, −1.547520175904541, −0.7928562721489236, 0, 0.7928562721489236, 1.547520175904541, 2.297956737697096, 2.914690135474110, 3.586339802171489, 4.238943282401341, 4.747399800313872, 5.457485798712125, 5.732342282658853, 6.244883137809137, 6.989514676440571, 7.639476476463520, 7.765729106305633, 8.306198296200274, 9.244011167752890, 9.784194546825599, 9.924872460723710, 10.37478607598978, 11.16275143890295, 11.71185026354661, 11.91259529806384, 12.34758421736065, 13.00424950468638, 13.38097358750198, 14.06260866986688

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