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 − 2·7-s + 9-s + 11-s − 4·13-s − 6·17-s + 4·19-s + 2·21-s − 6·23-s − 5·25-s − 27-s + 6·29-s − 8·31-s − 33-s − 10·37-s + 4·39-s + 6·41-s − 8·43-s + 6·47-s − 3·49-s + 6·51-s − 4·57-s + 8·61-s − 2·63-s + 4·67-s + 6·69-s − 6·71-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.755·7-s + 1/3·9-s + 0.301·11-s − 1.10·13-s − 1.45·17-s + 0.917·19-s + 0.436·21-s − 1.25·23-s − 25-s − 0.192·27-s + 1.11·29-s − 1.43·31-s − 0.174·33-s − 1.64·37-s + 0.640·39-s + 0.937·41-s − 1.21·43-s + 0.875·47-s − 3/7·49-s + 0.840·51-s − 0.529·57-s + 1.02·61-s − 0.251·63-s + 0.488·67-s + 0.722·69-s − 0.712·71-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 $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
11 \( 1 - T \)
good5 \( 1 + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
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 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
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

−19.75555390133291, −19.28023015983557, −18.13231914851664, −17.66134374595736, −16.90406525758345, −15.93208764480821, −15.66962501667217, −14.45710253656267, −13.69879028851631, −12.80670225998752, −12.05935089022954, −11.41314893400630, −10.28397817499554, −9.702896608623430, −8.767992898257258, −7.479996943637179, −6.733739842745446, −5.807715753214237, −4.750342018145241, −3.624850728814397, −2.125748707698263, 0, 2.125748707698263, 3.624850728814397, 4.750342018145241, 5.807715753214237, 6.733739842745446, 7.479996943637179, 8.767992898257258, 9.702896608623430, 10.28397817499554, 11.41314893400630, 12.05935089022954, 12.80670225998752, 13.69879028851631, 14.45710253656267, 15.66962501667217, 15.93208764480821, 16.90406525758345, 17.66134374595736, 18.13231914851664, 19.28023015983557, 19.75555390133291

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