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 − 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  =  0
Selberg data  =  $(2,\ 35739,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.084229574$
$L(\frac12)$  $\approx$  $1.084229574$
$L(\frac{3}{2})$   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

−14.72393643099934, −14.70687924372856, −13.76909702692379, −13.30883891383936, −12.76431322104590, −12.29638852316109, −12.09675164382289, −11.10572124277904, −11.06769748675466, −10.21515768683207, −9.377926069982859, −9.102612808731882, −8.359710444987837, −7.874429541842715, −7.440545767660799, −6.580401044067281, −6.185466129732394, −5.353600123068065, −4.760742005533437, −4.177011132503010, −3.763756264260263, −3.040865314691575, −2.805939622714208, −1.128517112650661, −0.4079444885027822, 0.4079444885027822, 1.128517112650661, 2.805939622714208, 3.040865314691575, 3.763756264260263, 4.177011132503010, 4.760742005533437, 5.353600123068065, 6.185466129732394, 6.580401044067281, 7.440545767660799, 7.874429541842715, 8.359710444987837, 9.102612808731882, 9.377926069982859, 10.21515768683207, 11.06769748675466, 11.10572124277904, 12.09675164382289, 12.29638852316109, 12.76431322104590, 13.30883891383936, 13.76909702692379, 14.70687924372856, 14.72393643099934

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