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  − 2·4-s + 2·7-s − 11-s + 13-s + 4·16-s − 3·17-s − 6·23-s − 5·25-s − 4·28-s − 8·31-s − 2·37-s + 6·41-s + 8·43-s + 2·44-s + 6·47-s − 3·49-s − 2·52-s + 9·53-s + 3·59-s − 10·61-s − 8·64-s + 10·67-s + 6·68-s − 3·71-s − 4·73-s − 2·77-s + 13·79-s + ⋯
L(s)  = 1  − 4-s + 0.755·7-s − 0.301·11-s + 0.277·13-s + 16-s − 0.727·17-s − 1.25·23-s − 25-s − 0.755·28-s − 1.43·31-s − 0.328·37-s + 0.937·41-s + 1.21·43-s + 0.301·44-s + 0.875·47-s − 3/7·49-s − 0.277·52-s + 1.23·53-s + 0.390·59-s − 1.28·61-s − 64-s + 1.22·67-s + 0.727·68-s − 0.356·71-s − 0.468·73-s − 0.227·77-s + 1.46·79-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 \( 1 + p T^{2} \)
5 \( 1 + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 + 3 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - 13 T + p T^{2} \)
83 \( 1 - 3 T + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−15.07510661487602, −14.55623115431067, −14.07221640828074, −13.71994699953911, −13.12382537827455, −12.67277224471404, −12.04657840597069, −11.54606388672360, −10.83705725701475, −10.49176869005428, −9.815248892651483, −9.164160791449335, −8.868679378885991, −8.177017272174408, −7.676894806018212, −7.302234152618492, −6.151207729589031, −5.864361959998415, −5.127791643380793, −4.620402723641414, −3.896007214320417, −3.634657201994255, −2.393182828239712, −1.893673468895786, −0.8843947177127464, 0, 0.8843947177127464, 1.893673468895786, 2.393182828239712, 3.634657201994255, 3.896007214320417, 4.620402723641414, 5.127791643380793, 5.864361959998415, 6.151207729589031, 7.302234152618492, 7.676894806018212, 8.177017272174408, 8.868679378885991, 9.164160791449335, 9.815248892651483, 10.49176869005428, 10.83705725701475, 11.54606388672360, 12.04657840597069, 12.67277224471404, 13.12382537827455, 13.71994699953911, 14.07221640828074, 14.55623115431067, 15.07510661487602

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