Properties

Degree 2
Conductor $ 2^{6} \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 − 5-s + 2·7-s − 2·9-s + 11-s − 4·13-s + 15-s − 2·17-s − 2·21-s + 23-s − 4·25-s + 5·27-s − 7·31-s − 33-s − 2·35-s − 3·37-s + 4·39-s − 8·41-s − 6·43-s + 2·45-s − 8·47-s − 3·49-s + 2·51-s + 6·53-s − 55-s + 5·59-s − 12·61-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 0.755·7-s − 2/3·9-s + 0.301·11-s − 1.10·13-s + 0.258·15-s − 0.485·17-s − 0.436·21-s + 0.208·23-s − 4/5·25-s + 0.962·27-s − 1.25·31-s − 0.174·33-s − 0.338·35-s − 0.493·37-s + 0.640·39-s − 1.24·41-s − 0.914·43-s + 0.298·45-s − 1.16·47-s − 3/7·49-s + 0.280·51-s + 0.824·53-s − 0.134·55-s + 0.650·59-s − 1.53·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(704\)    =    \(2^{6} \cdot 11\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{704} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 704,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;11\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 - T \)
good3 \( 1 + T + p T^{2} \)
5 \( 1 + T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 5 T + p T^{2} \)
61 \( 1 + 12 T + p T^{2} \)
67 \( 1 + 7 T + p T^{2} \)
71 \( 1 - 3 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
97 \( 1 + 7 T + p T^{2} \)
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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

−19.80875756221402, −19.40611247296194, −18.23287359417542, −17.76277484532720, −16.94363403156260, −16.56736210432717, −15.42262088776437, −14.80945524006568, −14.16472273811585, −13.20244372744763, −12.14105028387062, −11.68217477562548, −11.04905759572316, −10.14466147519939, −9.092146666164136, −8.248674172646585, −7.397014575004009, −6.463504409361631, −5.329920265916591, −4.696594207328282, −3.405798216950927, −1.945478870783472, 0, 1.945478870783472, 3.405798216950927, 4.696594207328282, 5.329920265916591, 6.463504409361631, 7.397014575004009, 8.248674172646585, 9.092146666164136, 10.14466147519939, 11.04905759572316, 11.68217477562548, 12.14105028387062, 13.20244372744763, 14.16472273811585, 14.80945524006568, 15.42262088776437, 16.56736210432717, 16.94363403156260, 17.76277484532720, 18.23287359417542, 19.40611247296194, 19.80875756221402

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