Properties

Degree 2
Conductor $ 2^{4} \cdot 3 \cdot 11 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 2·5-s + 9-s − 11-s + 2·13-s + 2·15-s + 6·17-s − 4·23-s − 25-s + 27-s + 2·29-s − 33-s − 10·37-s + 2·39-s + 6·41-s + 8·43-s + 2·45-s + 4·47-s − 7·49-s + 6·51-s − 6·53-s − 2·55-s + 12·59-s + 2·61-s + 4·65-s − 4·67-s − 4·69-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.894·5-s + 1/3·9-s − 0.301·11-s + 0.554·13-s + 0.516·15-s + 1.45·17-s − 0.834·23-s − 1/5·25-s + 0.192·27-s + 0.371·29-s − 0.174·33-s − 1.64·37-s + 0.320·39-s + 0.937·41-s + 1.21·43-s + 0.298·45-s + 0.583·47-s − 49-s + 0.840·51-s − 0.824·53-s − 0.269·55-s + 1.56·59-s + 0.256·61-s + 0.496·65-s − 0.488·67-s − 0.481·69-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  =  0
Selberg data  =  $(2,\ 528,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.048195850$
$L(\frac12)$  $\approx$  $2.048195850$
$L(\frac{3}{2})$   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 $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
11 \( 1 + T \)
good5 \( 1 - 2 T + p T^{2} \)
7 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
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 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
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

−19.23096449597753, −18.95987745832225, −17.80822616387445, −17.51974227060768, −16.26048462453699, −15.86409503253585, −14.65900961869832, −14.13646329986390, −13.47107543226462, −12.64556000294186, −11.79697917687588, −10.54048685043450, −9.981987073174720, −9.140836382652819, −8.212378907948041, −7.365876376407328, −6.129916819658788, −5.394985481453933, −3.982966802334034, −2.822553519430501, −1.546992350074350, 1.546992350074350, 2.822553519430501, 3.982966802334034, 5.394985481453933, 6.129916819658788, 7.365876376407328, 8.212378907948041, 9.140836382652819, 9.981987073174720, 10.54048685043450, 11.79697917687588, 12.64556000294186, 13.47107543226462, 14.13646329986390, 14.65900961869832, 15.86409503253585, 16.26048462453699, 17.51974227060768, 17.80822616387445, 18.95987745832225, 19.23096449597753

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