Properties

Degree 2
Conductor $ 3^{2} \cdot 11 \cdot 19^{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  + 2·2-s + 2·4-s − 4·7-s − 11-s + 13-s − 8·14-s − 4·16-s − 7·17-s − 2·22-s − 4·23-s − 5·25-s + 2·26-s − 8·28-s − 8·29-s − 2·31-s − 8·32-s − 14·34-s + 6·37-s + 6·43-s − 2·44-s − 8·46-s − 6·47-s + 9·49-s − 10·50-s + 2·52-s + 5·53-s − 16·58-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s − 1.51·7-s − 0.301·11-s + 0.277·13-s − 2.13·14-s − 16-s − 1.69·17-s − 0.426·22-s − 0.834·23-s − 25-s + 0.392·26-s − 1.51·28-s − 1.48·29-s − 0.359·31-s − 1.41·32-s − 2.40·34-s + 0.986·37-s + 0.914·43-s − 0.301·44-s − 1.17·46-s − 0.875·47-s + 9/7·49-s − 1.41·50-s + 0.277·52-s + 0.686·53-s − 2.10·58-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  =  0
Selberg data  =  $(2,\ 35739,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.8227325971$
$L(\frac12)$  $\approx$  $0.8227325971$
$L(\frac{3}{2})$   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 $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 + T \)
19 \( 1 \)
good2 \( 1 - p T + p T^{2} \)
5 \( 1 + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 + 7 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 5 T + p T^{2} \)
59 \( 1 + 9 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 15 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 13 T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 + 5 T + p T^{2} \)
97 \( 1 - 16 T + p T^{2} \)
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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

−14.96023859496082, −14.27258713731306, −13.70039807967730, −13.26439653413473, −12.93260059867969, −12.69229436106052, −11.70391941841791, −11.60244528403737, −10.80715313732012, −10.25436069920622, −9.492650585843006, −9.174159162210415, −8.587942356413403, −7.578665276087233, −7.178862101998033, −6.382299024477959, −5.999374788781869, −5.753242084974208, −4.734086556796747, −4.238658417024009, −3.709153011361210, −3.146141828861188, −2.447923671127191, −1.864087904717777, −0.2419081070836280, 0.2419081070836280, 1.864087904717777, 2.447923671127191, 3.146141828861188, 3.709153011361210, 4.238658417024009, 4.734086556796747, 5.753242084974208, 5.999374788781869, 6.382299024477959, 7.178862101998033, 7.578665276087233, 8.587942356413403, 9.174159162210415, 9.492650585843006, 10.25436069920622, 10.80715313732012, 11.60244528403737, 11.70391941841791, 12.69229436106052, 12.93260059867969, 13.26439653413473, 13.70039807967730, 14.27258713731306, 14.96023859496082

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