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 − 5-s − 3·7-s + 3·8-s + 10-s − 11-s + 6·13-s + 3·14-s − 16-s + 20-s + 22-s + 2·23-s − 4·25-s − 6·26-s + 3·28-s + 4·29-s − 4·31-s − 5·32-s + 3·35-s − 7·37-s − 3·40-s − 4·41-s − 4·43-s + 44-s − 2·46-s − 6·47-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s − 0.447·5-s − 1.13·7-s + 1.06·8-s + 0.316·10-s − 0.301·11-s + 1.66·13-s + 0.801·14-s − 1/4·16-s + 0.223·20-s + 0.213·22-s + 0.417·23-s − 4/5·25-s − 1.17·26-s + 0.566·28-s + 0.742·29-s − 0.718·31-s − 0.883·32-s + 0.507·35-s − 1.15·37-s − 0.474·40-s − 0.624·41-s − 0.609·43-s + 0.150·44-s − 0.294·46-s − 0.875·47-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  =  $(2,\ 35739,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.6124131621$
$L(\frac12)$  $\approx$  $0.6124131621$
$L(\frac{3}{2})$   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 \( 1 + T + p T^{2} \)
7 \( 1 + 3 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 15 T + p T^{2} \)
83 \( 1 - 17 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 - T + p T^{2} \)
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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

−15.15113928991086, −14.24157761700226, −13.71750273107825, −13.44765274858832, −12.78779728419490, −12.50759130648335, −11.58439649415785, −11.16806892550125, −10.51844815868013, −10.10673897258242, −9.548402128944840, −9.025897893470694, −8.422747183571228, −8.177127360312997, −7.426513605978292, −6.698181966306976, −6.386596599482596, −5.445658645057304, −5.036428008718672, −3.976185739331690, −3.706871006649308, −3.120397503346888, −2.022081850747118, −1.190420933121918, −0.3720706392916795, 0.3720706392916795, 1.190420933121918, 2.022081850747118, 3.120397503346888, 3.706871006649308, 3.976185739331690, 5.036428008718672, 5.445658645057304, 6.386596599482596, 6.698181966306976, 7.426513605978292, 8.177127360312997, 8.422747183571228, 9.025897893470694, 9.548402128944840, 10.10673897258242, 10.51844815868013, 11.16806892550125, 11.58439649415785, 12.50759130648335, 12.78779728419490, 13.44765274858832, 13.71750273107825, 14.24157761700226, 15.15113928991086

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