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-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  =  0
Selberg data  =  $(2,\ 35739,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $4.968086530$
$L(\frac12)$  $\approx$  $4.968086530$
$L(\frac{3}{2})$   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

−14.62299067994835, −14.48572299974736, −13.89752820080789, −13.30353185730809, −12.97886267469207, −12.63816374273673, −11.59129989349755, −11.31055458781733, −10.95168323086804, −10.05398347828465, −9.514395785516078, −9.066911574830044, −8.511947903675518, −8.014076086416335, −7.290889737611016, −6.513794197612639, −5.745436064533883, −5.636281114082134, −5.037165740810158, −4.189280863844745, −3.805301302171353, −3.116626840640993, −2.175096047320492, −1.464186151012192, −0.8087338487398218, 0.8087338487398218, 1.464186151012192, 2.175096047320492, 3.116626840640993, 3.805301302171353, 4.189280863844745, 5.037165740810158, 5.636281114082134, 5.745436064533883, 6.513794197612639, 7.290889737611016, 8.014076086416335, 8.511947903675518, 9.066911574830044, 9.514395785516078, 10.05398347828465, 10.95168323086804, 11.31055458781733, 11.59129989349755, 12.63816374273673, 12.97886267469207, 13.30353185730809, 13.89752820080789, 14.48572299974736, 14.62299067994835

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