Properties

Degree 2
Conductor $ 3^{2} \cdot 11 \cdot 19^{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 + 2·5-s + 4·7-s − 3·8-s + 2·10-s − 11-s + 2·13-s + 4·14-s − 16-s + 2·17-s − 2·20-s − 22-s − 8·23-s − 25-s + 2·26-s − 4·28-s − 6·29-s + 8·31-s + 5·32-s + 2·34-s + 8·35-s − 6·37-s − 6·40-s − 2·41-s + 44-s − 8·46-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s + 0.894·5-s + 1.51·7-s − 1.06·8-s + 0.632·10-s − 0.301·11-s + 0.554·13-s + 1.06·14-s − 1/4·16-s + 0.485·17-s − 0.447·20-s − 0.213·22-s − 1.66·23-s − 1/5·25-s + 0.392·26-s − 0.755·28-s − 1.11·29-s + 1.43·31-s + 0.883·32-s + 0.342·34-s + 1.35·35-s − 0.986·37-s − 0.948·40-s − 0.312·41-s + 0.150·44-s − 1.17·46-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 35739 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 35739 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(35739\)    =    \(3^{2} \cdot 11 \cdot 19^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{35739} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 35739,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;11,\;19\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;11,\;19\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 \)
11 \( 1 + T \)
19 \( 1 \)
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 \( 1 - 2 T + p T^{2} \)
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

−14.87280582876936, −14.60143429042677, −13.95683345791513, −13.75774324098098, −13.31482699323502, −12.67008567300838, −11.97366807389459, −11.70402136452086, −11.07553704495160, −10.28997230082097, −9.961096705289205, −9.371930621157807, −8.598600670427284, −8.231811927360942, −7.811009434922607, −6.894882164074324, −6.143633491475088, −5.593672057519809, −5.358588721798806, −4.593186481677012, −4.110568536525768, −3.447379229263192, −2.553436990240330, −1.842621549051463, −1.260216873152836, 0, 1.260216873152836, 1.842621549051463, 2.553436990240330, 3.447379229263192, 4.110568536525768, 4.593186481677012, 5.358588721798806, 5.593672057519809, 6.143633491475088, 6.894882164074324, 7.811009434922607, 8.231811927360942, 8.598600670427284, 9.371930621157807, 9.961096705289205, 10.28997230082097, 11.07553704495160, 11.70402136452086, 11.97366807389459, 12.67008567300838, 13.31482699323502, 13.75774324098098, 13.95683345791513, 14.60143429042677, 14.87280582876936

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