Properties

Degree 2
Conductor $ 2 \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  + 2-s + 3-s + 4-s − 4·5-s + 6-s − 2·7-s + 8-s + 9-s − 4·10-s + 11-s + 12-s + 4·13-s − 2·14-s − 4·15-s + 16-s − 2·17-s + 18-s − 4·20-s − 2·21-s + 22-s − 6·23-s + 24-s + 11·25-s + 4·26-s + 27-s − 2·28-s + 10·29-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.78·5-s + 0.408·6-s − 0.755·7-s + 0.353·8-s + 1/3·9-s − 1.26·10-s + 0.301·11-s + 0.288·12-s + 1.10·13-s − 0.534·14-s − 1.03·15-s + 1/4·16-s − 0.485·17-s + 0.235·18-s − 0.894·20-s − 0.436·21-s + 0.213·22-s − 1.25·23-s + 0.204·24-s + 11/5·25-s + 0.784·26-s + 0.192·27-s − 0.377·28-s + 1.85·29-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 66 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 66 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(66\)    =    \(2 \cdot 3 \cdot 11\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{66} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 66,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.191614815$
$L(\frac12)$  $\approx$  $1.191614815$
$L(\frac{3}{2})$   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 \( 1 - T \)
3 \( 1 - T \)
11 \( 1 - T \)
good5 \( 1 + 4 T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
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

−19.78014225188802, −19.43482598444865, −18.22009854578069, −16.18314677756459, −15.92612139416214, −14.93063692211966, −13.79927171797323, −12.63981185671263, −11.77772233494395, −10.63196865714990, −8.822664775845247, −7.723393030532525, −6.469171817054118, −4.258421785479845, −3.348334362632695, 3.348334362632695, 4.258421785479845, 6.469171817054118, 7.723393030532525, 8.822664775845247, 10.63196865714990, 11.77772233494395, 12.63981185671263, 13.79927171797323, 14.93063692211966, 15.92612139416214, 16.18314677756459, 18.22009854578069, 19.43482598444865, 19.78014225188802

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