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-s − 4-s + 4·5-s − 2·7-s + 3·8-s − 4·10-s + 11-s + 2·13-s + 2·14-s − 16-s − 2·17-s − 4·20-s − 22-s − 4·23-s + 11·25-s − 2·26-s + 2·28-s − 6·29-s − 4·31-s − 5·32-s + 2·34-s − 8·35-s + 6·37-s + 12·40-s − 10·41-s + 6·43-s − 44-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s + 1.78·5-s − 0.755·7-s + 1.06·8-s − 1.26·10-s + 0.301·11-s + 0.554·13-s + 0.534·14-s − 1/4·16-s − 0.485·17-s − 0.894·20-s − 0.213·22-s − 0.834·23-s + 11/5·25-s − 0.392·26-s + 0.377·28-s − 1.11·29-s − 0.718·31-s − 0.883·32-s + 0.342·34-s − 1.35·35-s + 0.986·37-s + 1.89·40-s − 1.56·41-s + 0.914·43-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  =  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 \( 1 + T + p T^{2} \)
5 \( 1 - 4 T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−15.10513151015001, −14.53842216108540, −13.97077829562711, −13.56988737950937, −13.22520355104426, −12.77893306857432, −12.18212297361010, −11.19083820732738, −10.79772581848862, −10.16844504183719, −9.713852666267388, −9.472970785371913, −8.768609309523580, −8.617080873975320, −7.561042592122057, −7.106616181730193, −6.233906106378191, −6.011467114136856, −5.381734867369248, −4.656652411778674, −3.934476019348819, −3.238925169084478, −2.267688698910028, −1.772284425708589, −1.032749093783487, 0, 1.032749093783487, 1.772284425708589, 2.267688698910028, 3.238925169084478, 3.934476019348819, 4.656652411778674, 5.381734867369248, 6.011467114136856, 6.233906106378191, 7.106616181730193, 7.561042592122057, 8.617080873975320, 8.768609309523580, 9.472970785371913, 9.713852666267388, 10.16844504183719, 10.79772581848862, 11.19083820732738, 12.18212297361010, 12.77893306857432, 13.22520355104426, 13.56988737950937, 13.97077829562711, 14.53842216108540, 15.10513151015001

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