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  − 2·2-s + 2·4-s − 5-s − 2·7-s + 2·10-s − 11-s − 4·13-s + 4·14-s − 4·16-s + 2·17-s − 2·20-s + 2·22-s + 23-s − 4·25-s + 8·26-s − 4·28-s − 7·31-s + 8·32-s − 4·34-s + 2·35-s − 3·37-s − 8·41-s − 6·43-s − 2·44-s − 2·46-s − 8·47-s − 3·49-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s − 0.447·5-s − 0.755·7-s + 0.632·10-s − 0.301·11-s − 1.10·13-s + 1.06·14-s − 16-s + 0.485·17-s − 0.447·20-s + 0.426·22-s + 0.208·23-s − 4/5·25-s + 1.56·26-s − 0.755·28-s − 1.25·31-s + 1.41·32-s − 0.685·34-s + 0.338·35-s − 0.493·37-s − 1.24·41-s − 0.914·43-s − 0.301·44-s − 0.294·46-s − 1.16·47-s − 3/7·49-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 $F_p(T)$ 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 \( 1 + T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 5 T + p T^{2} \)
61 \( 1 - 12 T + p T^{2} \)
67 \( 1 - 7 T + p T^{2} \)
71 \( 1 + 3 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
97 \( 1 - 7 T + p T^{2} \)
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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

−15.43646541274814, −14.69755339750472, −14.35754310137328, −13.41311754030463, −13.13225285954066, −12.43851426360170, −11.88883480462969, −11.39330458973174, −10.81839215255947, −10.13224297092343, −9.811197593105814, −9.479024141185336, −8.732730655642576, −8.214596534850173, −7.735563513891799, −7.171157551694397, −6.776546032852449, −6.058947876590397, −5.096907940657888, −4.806742581058586, −3.629240729406667, −3.332804903741540, −2.243264751143302, −1.787876874565955, −0.6226040801845123, 0, 0.6226040801845123, 1.787876874565955, 2.243264751143302, 3.332804903741540, 3.629240729406667, 4.806742581058586, 5.096907940657888, 6.058947876590397, 6.776546032852449, 7.171157551694397, 7.735563513891799, 8.214596534850173, 8.732730655642576, 9.479024141185336, 9.811197593105814, 10.13224297092343, 10.81839215255947, 11.39330458973174, 11.88883480462969, 12.43851426360170, 13.13225285954066, 13.41311754030463, 14.35754310137328, 14.69755339750472, 15.43646541274814

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