Properties

Degree 2
Conductor $ 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  + 2-s − 3-s − 4-s − 2·5-s − 6-s + 4·7-s − 3·8-s + 9-s − 2·10-s + 11-s + 12-s − 2·13-s + 4·14-s + 2·15-s − 16-s − 2·17-s + 18-s + 2·20-s − 4·21-s + 22-s + 8·23-s + 3·24-s − 25-s − 2·26-s − 27-s − 4·28-s − 6·29-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.894·5-s − 0.408·6-s + 1.51·7-s − 1.06·8-s + 1/3·9-s − 0.632·10-s + 0.301·11-s + 0.288·12-s − 0.554·13-s + 1.06·14-s + 0.516·15-s − 1/4·16-s − 0.485·17-s + 0.235·18-s + 0.447·20-s − 0.872·21-s + 0.213·22-s + 1.66·23-s + 0.612·24-s − 1/5·25-s − 0.392·26-s − 0.192·27-s − 0.755·28-s − 1.11·29-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(33\)    =    \(3 \cdot 11\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{33} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 33,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.7473391477$
$L(\frac12)$  $\approx$  $0.7473391477$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

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

−18.88037562185510, −17.86377594524811, −16.90261044466837, −15.19333100007082, −14.59050172699885, −13.10139310227399, −11.86897475867739, −11.06987917382067, −8.975261543279014, −7.469958161844614, −5.316224825638298, −4.204079032833299, 4.204079032833299, 5.316224825638298, 7.469958161844614, 8.975261543279014, 11.06987917382067, 11.86897475867739, 13.10139310227399, 14.59050172699885, 15.19333100007082, 16.90261044466837, 17.86377594524811, 18.88037562185510

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