Properties

Degree 2
Conductor $ 2^{4} \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 − 4·7-s − 3·9-s + 2·13-s − 2·17-s + 4·19-s − 4·23-s + 25-s + 2·29-s + 8·31-s − 4·35-s + 6·37-s + 6·41-s − 8·43-s − 3·45-s − 4·47-s + 9·49-s + 6·53-s + 4·59-s + 2·61-s + 12·63-s + 2·65-s − 8·67-s + 6·73-s + 9·81-s − 16·83-s − 2·85-s + ⋯
L(s)  = 1  + 0.447·5-s − 1.51·7-s − 9-s + 0.554·13-s − 0.485·17-s + 0.917·19-s − 0.834·23-s + 1/5·25-s + 0.371·29-s + 1.43·31-s − 0.676·35-s + 0.986·37-s + 0.937·41-s − 1.21·43-s − 0.447·45-s − 0.583·47-s + 9/7·49-s + 0.824·53-s + 0.520·59-s + 0.256·61-s + 1.51·63-s + 0.248·65-s − 0.977·67-s + 0.702·73-s + 81-s − 1.75·83-s − 0.216·85-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(9680\)    =    \(2^{4} \cdot 5 \cdot 11^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{9680} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 9680,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5,\;11\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;5,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 - T \)
11 \( 1 \)
good3 \( 1 + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
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

−16.88825050419975, −16.31938108076882, −15.99713412845822, −15.34054984436578, −14.65404157049159, −13.88986402864873, −13.56701361424807, −13.07851944135047, −12.32034140993002, −11.77183619555679, −11.17701898816538, −10.41541491817067, −9.688162585210371, −9.549057509548747, −8.563879011661794, −8.223522789927761, −7.173335215057393, −6.513025907569490, −6.028385690346504, −5.541532793905458, −4.532439017351402, −3.665741718656563, −2.944678740374268, −2.448213597443158, −1.095980942596233, 0, 1.095980942596233, 2.448213597443158, 2.944678740374268, 3.665741718656563, 4.532439017351402, 5.541532793905458, 6.028385690346504, 6.513025907569490, 7.173335215057393, 8.223522789927761, 8.563879011661794, 9.549057509548747, 9.688162585210371, 10.41541491817067, 11.17701898816538, 11.77183619555679, 12.32034140993002, 13.07851944135047, 13.56701361424807, 13.88986402864873, 14.65404157049159, 15.34054984436578, 15.99713412845822, 16.31938108076882, 16.88825050419975

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