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 1

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  =  1
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

−15.03153084770906, −14.48409207347117, −13.92600308696397, −13.39848855583419, −12.75140835517943, −12.18874093047896, −11.92458197653723, −11.39178725818803, −10.73441843574697, −10.27416068909654, −9.761960175574286, −9.091336985370322, −8.396457121526826, −8.093429903651314, −7.507889577021383, −6.866730667339226, −6.592111542403195, −5.416309792359217, −5.206516687448500, −4.332794000965034, −4.088451252456579, −3.213037443607844, −2.375058756828255, −2.035088477538087, −0.7479756119964155, 0, 0.7479756119964155, 2.035088477538087, 2.375058756828255, 3.213037443607844, 4.088451252456579, 4.332794000965034, 5.206516687448500, 5.416309792359217, 6.592111542403195, 6.866730667339226, 7.507889577021383, 8.093429903651314, 8.396457121526826, 9.091336985370322, 9.761960175574286, 10.27416068909654, 10.73441843574697, 11.39178725818803, 11.92458197653723, 12.18874093047896, 12.75140835517943, 13.39848855583419, 13.92600308696397, 14.48409207347117, 15.03153084770906

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