Properties

Degree 2
Conductor $ 2^{4} \cdot 3 \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 − 2·5-s − 4·7-s + 9-s − 11-s − 2·13-s − 2·15-s − 2·17-s − 4·21-s − 8·23-s − 25-s + 27-s − 6·29-s + 8·31-s − 33-s + 8·35-s + 6·37-s − 2·39-s − 2·41-s − 2·45-s − 8·47-s + 9·49-s − 2·51-s + 6·53-s + 2·55-s + 4·59-s + 6·61-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.894·5-s − 1.51·7-s + 1/3·9-s − 0.301·11-s − 0.554·13-s − 0.516·15-s − 0.485·17-s − 0.872·21-s − 1.66·23-s − 1/5·25-s + 0.192·27-s − 1.11·29-s + 1.43·31-s − 0.174·33-s + 1.35·35-s + 0.986·37-s − 0.320·39-s − 0.312·41-s − 0.298·45-s − 1.16·47-s + 9/7·49-s − 0.280·51-s + 0.824·53-s + 0.269·55-s + 0.520·59-s + 0.768·61-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(528\)    =    \(2^{4} \cdot 3 \cdot 11\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{528} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 528,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;11\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;11\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
3 \( 1 - T \)
11 \( 1 + T \)
good5 \( 1 + 2 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
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

−19.82341111693782, −19.26014969259556, −18.63236855009406, −17.68319177837966, −16.56777151171542, −15.97238256126340, −15.44941819839620, −14.63834793095151, −13.58422329719866, −12.99335932950097, −12.17027432564283, −11.41048898950594, −10.05246880525653, −9.723139098205725, −8.548353092044890, −7.757207820232251, −6.879282631439092, −5.907237385521307, −4.354610893746684, −3.530741785587904, −2.452500231125232, 0, 2.452500231125232, 3.530741785587904, 4.354610893746684, 5.907237385521307, 6.879282631439092, 7.757207820232251, 8.548353092044890, 9.723139098205725, 10.05246880525653, 11.41048898950594, 12.17027432564283, 12.99335932950097, 13.58422329719866, 14.63834793095151, 15.44941819839620, 15.97238256126340, 16.56777151171542, 17.68319177837966, 18.63236855009406, 19.26014969259556, 19.82341111693782

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