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  − 2·4-s − 4·5-s + 2·7-s + 11-s − 13-s + 4·16-s + 3·17-s + 8·20-s − 2·23-s + 11·25-s − 4·28-s − 4·29-s + 4·31-s − 8·35-s − 2·37-s + 6·41-s − 4·43-s − 2·44-s − 6·47-s − 3·49-s + 2·52-s + 53-s − 4·55-s + 11·59-s + 10·61-s − 8·64-s + 4·65-s + ⋯
L(s)  = 1  − 4-s − 1.78·5-s + 0.755·7-s + 0.301·11-s − 0.277·13-s + 16-s + 0.727·17-s + 1.78·20-s − 0.417·23-s + 11/5·25-s − 0.755·28-s − 0.742·29-s + 0.718·31-s − 1.35·35-s − 0.328·37-s + 0.937·41-s − 0.609·43-s − 0.301·44-s − 0.875·47-s − 3/7·49-s + 0.277·52-s + 0.137·53-s − 0.539·55-s + 1.43·59-s + 1.28·61-s − 64-s + 0.496·65-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  =  $(2,\ 35739,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.098227910$
$L(\frac12)$  $\approx$  $1.098227910$
$L(\frac{3}{2})$   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 \( 1 + p T^{2} \)
5 \( 1 + 4 T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - T + p T^{2} \)
59 \( 1 - 11 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 6 T + p T^{2} \)
71 \( 1 + 3 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 - 11 T + p T^{2} \)
83 \( 1 - 13 T + p T^{2} \)
89 \( 1 + 9 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−14.85253344472020, −14.51699988174757, −14.04199292892325, −13.30656503699264, −12.73616579760375, −12.23221878301630, −11.81991870338001, −11.29570395596966, −10.87997438690347, −9.969567940192457, −9.709638937823295, −8.732600802967118, −8.466902313286008, −7.933565047901953, −7.551996138111253, −6.951072105272547, −6.083526950385420, −5.246969638857433, −4.808304502939040, −4.293300189764415, −3.616220253421320, −3.370285188638210, −2.200146320110244, −1.109636628735825, −0.4760028863335252, 0.4760028863335252, 1.109636628735825, 2.200146320110244, 3.370285188638210, 3.616220253421320, 4.293300189764415, 4.808304502939040, 5.246969638857433, 6.083526950385420, 6.951072105272547, 7.551996138111253, 7.933565047901953, 8.466902313286008, 8.732600802967118, 9.709638937823295, 9.969567940192457, 10.87997438690347, 11.29570395596966, 11.81991870338001, 12.23221878301630, 12.73616579760375, 13.30656503699264, 14.04199292892325, 14.51699988174757, 14.85253344472020

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