Properties

Degree 2
Conductor $ 3^{2} \cdot 11 \cdot 19^{2} $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 2

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  =  2
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 \( 1 + p T^{2} \)
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

−15.75091220345416, −15.35629769201092, −14.30360198202597, −13.82450486907951, −13.29489455108111, −12.86184214875695, −12.22363383170587, −11.61821055515322, −11.05299876598451, −10.42808444934094, −9.933126204851828, −9.727054239966961, −9.049380454551163, −8.533957185736317, −8.129932012826753, −7.323557881397419, −6.844031990012381, −6.413368241603369, −5.837447583561406, −4.839332237688711, −4.241077879296667, −3.513255447226527, −2.571401848993892, −2.232889004579804, −1.194342433940191, 0, 0, 1.194342433940191, 2.232889004579804, 2.571401848993892, 3.513255447226527, 4.241077879296667, 4.839332237688711, 5.837447583561406, 6.413368241603369, 6.844031990012381, 7.323557881397419, 8.129932012826753, 8.533957185736317, 9.049380454551163, 9.727054239966961, 9.933126204851828, 10.42808444934094, 11.05299876598451, 11.61821055515322, 12.22363383170587, 12.86184214875695, 13.29489455108111, 13.82450486907951, 14.30360198202597, 15.35629769201092, 15.75091220345416

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