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 + 2·5-s + 2·7-s − 11-s + 5·13-s + 4·16-s − 3·17-s − 4·20-s + 6·23-s − 25-s − 4·28-s − 10·29-s + 10·31-s + 4·35-s − 10·41-s − 6·43-s + 2·44-s − 2·47-s − 3·49-s − 10·52-s + 5·53-s − 2·55-s − 5·59-s − 8·64-s + 10·65-s + 10·67-s + 6·68-s + ⋯
L(s)  = 1  − 4-s + 0.894·5-s + 0.755·7-s − 0.301·11-s + 1.38·13-s + 16-s − 0.727·17-s − 0.894·20-s + 1.25·23-s − 1/5·25-s − 0.755·28-s − 1.85·29-s + 1.79·31-s + 0.676·35-s − 1.56·41-s − 0.914·43-s + 0.301·44-s − 0.291·47-s − 3/7·49-s − 1.38·52-s + 0.686·53-s − 0.269·55-s − 0.650·59-s − 64-s + 1.24·65-s + 1.22·67-s + 0.727·68-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 \( 1 - 2 T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 - 5 T + p T^{2} \)
59 \( 1 + 5 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 + 15 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 5 T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 + 5 T + p T^{2} \)
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

−15.20000186506632, −14.61735518601448, −13.91428042642458, −13.58809721849215, −13.22381794424949, −12.90937008064049, −12.01840872316523, −11.35105545378201, −11.02011109431793, −10.30848739950714, −9.858214407594245, −9.262615814370497, −8.715658243207720, −8.365140018920674, −7.809644016288665, −6.963133385854783, −6.301251266320398, −5.793177744122787, −5.030109457901862, −4.873000405281182, −3.924241000999192, −3.439646382175132, −2.520121546672903, −1.634658197562532, −1.165494150200080, 0, 1.165494150200080, 1.634658197562532, 2.520121546672903, 3.439646382175132, 3.924241000999192, 4.873000405281182, 5.030109457901862, 5.793177744122787, 6.301251266320398, 6.963133385854783, 7.809644016288665, 8.365140018920674, 8.715658243207720, 9.262615814370497, 9.858214407594245, 10.30848739950714, 11.02011109431793, 11.35105545378201, 12.01840872316523, 12.90937008064049, 13.22381794424949, 13.58809721849215, 13.91428042642458, 14.61735518601448, 15.20000186506632

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