Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{2} \cdot 5 \cdot 11^{2} $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s − 2·7-s − 2·13-s − 6·17-s + 4·19-s − 6·23-s + 25-s + 6·29-s − 4·31-s − 2·35-s + 2·37-s + 6·41-s + 10·43-s + 6·47-s − 3·49-s + 6·53-s − 12·59-s − 2·61-s − 2·65-s + 2·67-s + 12·71-s − 2·73-s − 8·79-s + 6·83-s − 6·85-s + 6·89-s + 4·91-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.755·7-s − 0.554·13-s − 1.45·17-s + 0.917·19-s − 1.25·23-s + 1/5·25-s + 1.11·29-s − 0.718·31-s − 0.338·35-s + 0.328·37-s + 0.937·41-s + 1.52·43-s + 0.875·47-s − 3/7·49-s + 0.824·53-s − 1.56·59-s − 0.256·61-s − 0.248·65-s + 0.244·67-s + 1.42·71-s − 0.234·73-s − 0.900·79-s + 0.658·83-s − 0.650·85-s + 0.635·89-s + 0.419·91-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21780 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21780 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(21780\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 11^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{21780} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 21780,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;11\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 - T \)
11 \( 1 \)
good7 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
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 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
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

−15.91053990007987, −15.42718898683764, −14.68137790561565, −13.97886570928675, −13.83554966501341, −13.09190740342869, −12.52885682591703, −12.18772160061789, −11.38745522362678, −10.88390056246560, −10.19173935643563, −9.781093817554055, −9.108607155862283, −8.844836097533042, −7.802350900167554, −7.434532923981769, −6.621864696153497, −6.184018143160976, −5.610898372444905, −4.783000451526619, −4.212513262415596, −3.449131971724261, −2.566602231436570, −2.179483921647313, −1.010923854357220, 0, 1.010923854357220, 2.179483921647313, 2.566602231436570, 3.449131971724261, 4.212513262415596, 4.783000451526619, 5.610898372444905, 6.184018143160976, 6.621864696153497, 7.434532923981769, 7.802350900167554, 8.844836097533042, 9.108607155862283, 9.781093817554055, 10.19173935643563, 10.88390056246560, 11.38745522362678, 12.18772160061789, 12.52885682591703, 13.09190740342869, 13.83554966501341, 13.97886570928675, 14.68137790561565, 15.42718898683764, 15.91053990007987

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