Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 11^{2} $
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 + 4-s − 2·7-s − 8-s + 4·13-s + 2·14-s + 16-s − 6·17-s + 4·19-s − 6·23-s − 5·25-s − 4·26-s − 2·28-s + 6·29-s + 8·31-s − 32-s + 6·34-s − 10·37-s − 4·38-s + 6·41-s − 8·43-s + 6·46-s + 6·47-s − 3·49-s + 5·50-s + 4·52-s + 2·56-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.755·7-s − 0.353·8-s + 1.10·13-s + 0.534·14-s + 1/4·16-s − 1.45·17-s + 0.917·19-s − 1.25·23-s − 25-s − 0.784·26-s − 0.377·28-s + 1.11·29-s + 1.43·31-s − 0.176·32-s + 1.02·34-s − 1.64·37-s − 0.648·38-s + 0.937·41-s − 1.21·43-s + 0.884·46-s + 0.875·47-s − 3/7·49-s + 0.707·50-s + 0.554·52-s + 0.267·56-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2178 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2178 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(2178\)    =    \(2 \cdot 3^{2} \cdot 11^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{2178} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 2178,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 \( 1 + T \)
3 \( 1 \)
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

−19.66120149976102, −18.87803155024038, −18.20836224736652, −17.68206905150901, −17.17484395818710, −16.05371627178224, −15.82066099090064, −15.49808851261466, −14.19138722777604, −13.64113561750800, −13.08725930011775, −11.95936140841314, −11.70904779252948, −10.65388729663943, −10.16022201070390, −9.411814401734375, −8.672134764185654, −8.107075144714635, −7.118128042229707, −6.374597031871440, −5.863762317319803, −4.541320761392206, −3.593006997024930, −2.639154826245720, −1.456689189862392, 0, 1.456689189862392, 2.639154826245720, 3.593006997024930, 4.541320761392206, 5.863762317319803, 6.374597031871440, 7.118128042229707, 8.107075144714635, 8.672134764185654, 9.411814401734375, 10.16022201070390, 10.65388729663943, 11.70904779252948, 11.95936140841314, 13.08725930011775, 13.64113561750800, 14.19138722777604, 15.49808851261466, 15.82066099090064, 16.05371627178224, 17.17484395818710, 17.68206905150901, 18.20836224736652, 18.87803155024038, 19.66120149976102

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