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 + 3·7-s + 3·8-s − 2·10-s + 11-s − 6·13-s − 3·14-s − 16-s + 3·17-s − 2·20-s − 22-s + 5·23-s − 25-s + 6·26-s − 3·28-s + 7·29-s + 8·31-s − 5·32-s − 3·34-s + 6·35-s − 10·37-s + 6·40-s + 2·41-s + 11·43-s − 44-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s + 0.894·5-s + 1.13·7-s + 1.06·8-s − 0.632·10-s + 0.301·11-s − 1.66·13-s − 0.801·14-s − 1/4·16-s + 0.727·17-s − 0.447·20-s − 0.213·22-s + 1.04·23-s − 1/5·25-s + 1.17·26-s − 0.566·28-s + 1.29·29-s + 1.43·31-s − 0.883·32-s − 0.514·34-s + 1.01·35-s − 1.64·37-s + 0.948·40-s + 0.312·41-s + 1.67·43-s − 0.150·44-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 \( 1 + T + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
7 \( 1 - 3 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
23 \( 1 - 5 T + p T^{2} \)
29 \( 1 - 7 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 11 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 + 3 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 10 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
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

−15.03225049892979, −14.40125766106710, −14.29478779035891, −13.74372612945224, −13.20641065372450, −12.43140446053210, −12.09827330704331, −11.45528025253878, −10.73787250335184, −10.17574147416708, −9.943436386719278, −9.305479220640126, −8.841072322890902, −8.214029203742480, −7.750351229582568, −7.194765350158585, −6.587238248832940, −5.661368519472142, −5.175888246452797, −4.662188704293023, −4.253733182781673, −3.039227736417744, −2.456681785982814, −1.516641404247767, −1.175877863225698, 0, 1.175877863225698, 1.516641404247767, 2.456681785982814, 3.039227736417744, 4.253733182781673, 4.662188704293023, 5.175888246452797, 5.661368519472142, 6.587238248832940, 7.194765350158585, 7.750351229582568, 8.214029203742480, 8.841072322890902, 9.305479220640126, 9.943436386719278, 10.17574147416708, 10.73787250335184, 11.45528025253878, 12.09827330704331, 12.43140446053210, 13.20641065372450, 13.74372612945224, 14.29478779035891, 14.40125766106710, 15.03225049892979

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