Properties

Degree 2
Conductor $ 3^{2} \cdot 11 \cdot 19^{2} $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 2

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  =  2
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

−15.08108849017373, −14.82383785803327, −14.64464358051832, −13.85968252368726, −13.47294539169470, −12.70693533434723, −12.28327321649341, −11.80313763839510, −11.42221824635763, −10.62736015545344, −10.07545143108216, −9.543324890059704, −9.068231957261125, −8.390754563255794, −7.799743791808884, −7.589081260679571, −6.978157985614531, −5.911589796961541, −5.356908605672917, −4.941611894286347, −4.185796882856437, −3.781575349351974, −3.203753752336370, −2.023071304169786, −1.470913199991305, 0, 0, 1.470913199991305, 2.023071304169786, 3.203753752336370, 3.781575349351974, 4.185796882856437, 4.941611894286347, 5.356908605672917, 5.911589796961541, 6.978157985614531, 7.589081260679571, 7.799743791808884, 8.390754563255794, 9.068231957261125, 9.543324890059704, 10.07545143108216, 10.62736015545344, 11.42221824635763, 11.80313763839510, 12.28327321649341, 12.70693533434723, 13.47294539169470, 13.85968252368726, 14.64464358051832, 14.82383785803327, 15.08108849017373

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