Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s − 7-s + 11-s − 6·13-s − 2·17-s + 4·19-s − 25-s − 2·29-s − 8·31-s + 2·35-s − 6·37-s − 10·41-s − 4·43-s − 8·47-s + 49-s + 6·53-s − 2·55-s − 4·59-s + 10·61-s + 12·65-s − 12·67-s + 2·73-s − 77-s − 16·79-s − 4·83-s + 4·85-s − 18·89-s + ⋯
L(s)  = 1  − 0.894·5-s − 0.377·7-s + 0.301·11-s − 1.66·13-s − 0.485·17-s + 0.917·19-s − 1/5·25-s − 0.371·29-s − 1.43·31-s + 0.338·35-s − 0.986·37-s − 1.56·41-s − 0.609·43-s − 1.16·47-s + 1/7·49-s + 0.824·53-s − 0.269·55-s − 0.520·59-s + 1.28·61-s + 1.48·65-s − 1.46·67-s + 0.234·73-s − 0.113·77-s − 1.80·79-s − 0.439·83-s + 0.433·85-s − 1.90·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 44352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(44352\)    =    \(2^{6} \cdot 3^{2} \cdot 7 \cdot 11\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{44352} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  2
Selberg data  =  $(2,\ 44352,\ (\ :1/2),\ 1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 \( 1 \)
7 \( 1 + T \)
11 \( 1 - T \)
good5 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
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

−15.31255740619993, −14.61429140677682, −14.34206492428631, −13.55734961876357, −13.11853237379457, −12.45944088163424, −12.09151653041275, −11.53123119408761, −11.29406261683786, −10.35655581350758, −9.946264036372715, −9.497441719093682, −8.805668579897763, −8.350693265528723, −7.583263270589660, −7.131623057885274, −6.934018053037100, −5.977959703130231, −5.278000924410227, −4.869995592498266, −4.115215677344699, −3.525842316449838, −3.012840999200863, −2.153277269614215, −1.436935358021841, 0, 0, 1.436935358021841, 2.153277269614215, 3.012840999200863, 3.525842316449838, 4.115215677344699, 4.869995592498266, 5.278000924410227, 5.977959703130231, 6.934018053037100, 7.131623057885274, 7.583263270589660, 8.350693265528723, 8.805668579897763, 9.497441719093682, 9.946264036372715, 10.35655581350758, 11.29406261683786, 11.53123119408761, 12.09151653041275, 12.45944088163424, 13.11853237379457, 13.55734961876357, 14.34206492428631, 14.61429140677682, 15.31255740619993

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