Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 11^{2} $
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 + 6-s − 2·7-s + 8-s + 9-s + 12-s + 4·13-s − 2·14-s + 16-s + 6·17-s + 18-s + 4·19-s − 2·21-s + 6·23-s + 24-s − 5·25-s + 4·26-s + 27-s − 2·28-s − 6·29-s + 8·31-s + 32-s + 6·34-s + 36-s − 10·37-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.755·7-s + 0.353·8-s + 1/3·9-s + 0.288·12-s + 1.10·13-s − 0.534·14-s + 1/4·16-s + 1.45·17-s + 0.235·18-s + 0.917·19-s − 0.436·21-s + 1.25·23-s + 0.204·24-s − 25-s + 0.784·26-s + 0.192·27-s − 0.377·28-s − 1.11·29-s + 1.43·31-s + 0.176·32-s + 1.02·34-s + 1/6·36-s − 1.64·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 726 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 726 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(726\)    =    \(2 \cdot 3 \cdot 11^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{726} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 726,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.828514425$
$L(\frac12)$  $\approx$  $2.828514425$
$L(\frac{3}{2})$   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 \( 1 - T \)
3 \( 1 - T \)
11 \( 1 \)
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

−19.44712763210866, −18.98247438383383, −18.27141527518289, −17.10572064087200, −16.41029744471533, −15.66240175731110, −15.15424929527263, −14.17559338583218, −13.56671531824004, −13.04405592720258, −12.10143921198451, −11.41966127658624, −10.27984892054234, −9.670414706004005, −8.636957582587328, −7.722966860170220, −6.824793417263956, −5.903834801086031, −4.965023564379852, −3.479006231310799, −3.241396009548310, −1.498767544255660, 1.498767544255660, 3.241396009548310, 3.479006231310799, 4.965023564379852, 5.903834801086031, 6.824793417263956, 7.722966860170220, 8.636957582587328, 9.670414706004005, 10.27984892054234, 11.41966127658624, 12.10143921198451, 13.04405592720258, 13.56671531824004, 14.17559338583218, 15.15424929527263, 15.66240175731110, 16.41029744471533, 17.10572064087200, 18.27141527518289, 18.98247438383383, 19.44712763210866

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