Properties

Degree 2
Conductor $ 3 \cdot 5^{2} \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  − 2-s + 3-s − 4-s − 6-s − 4·7-s + 3·8-s + 9-s + 11-s − 12-s + 2·13-s + 4·14-s − 16-s + 2·17-s − 18-s − 4·21-s − 22-s − 8·23-s + 3·24-s − 2·26-s + 27-s + 4·28-s − 6·29-s − 8·31-s − 5·32-s + 33-s − 2·34-s − 36-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.408·6-s − 1.51·7-s + 1.06·8-s + 1/3·9-s + 0.301·11-s − 0.288·12-s + 0.554·13-s + 1.06·14-s − 1/4·16-s + 0.485·17-s − 0.235·18-s − 0.872·21-s − 0.213·22-s − 1.66·23-s + 0.612·24-s − 0.392·26-s + 0.192·27-s + 0.755·28-s − 1.11·29-s − 1.43·31-s − 0.883·32-s + 0.174·33-s − 0.342·34-s − 1/6·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{825} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 825,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;11\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;5,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 - T \)
5 \( 1 \)
11 \( 1 - T \)
good2 \( 1 + T + p T^{2} \)
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

−19.90537482596308, −19.59972220535646, −18.69362789989506, −18.46793981499858, −17.51326894660938, −16.56758112248558, −16.23531903606972, −15.39841493928878, −14.31460917596303, −13.79935175775781, −12.95111868942299, −12.50930088312825, −11.23206328697742, −10.17772545901373, −9.705159010656125, −9.083856298436523, −8.281293500899591, −7.433988970480008, −6.488360086473925, −5.458073030969560, −3.959493293324536, −3.415922779412649, −1.765590175245489, 0, 1.765590175245489, 3.415922779412649, 3.959493293324536, 5.458073030969560, 6.488360086473925, 7.433988970480008, 8.281293500899591, 9.083856298436523, 9.705159010656125, 10.17772545901373, 11.23206328697742, 12.50930088312825, 12.95111868942299, 13.79935175775781, 14.31460917596303, 15.39841493928878, 16.23531903606972, 16.56758112248558, 17.51326894660938, 18.46793981499858, 18.69362789989506, 19.59972220535646, 19.90537482596308

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