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  − 2·4-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  =  0
Selberg data  =  $(2,\ 35739,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.195914830$
$L(\frac12)$  $\approx$  $2.195914830$
$L(\frac{3}{2})$   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

−14.90113894035141, −14.19036695123450, −13.87529323571655, −13.67793043359234, −12.61300981261088, −12.41201934984530, −11.70022466657102, −11.38642806180634, −10.69931737977166, −10.04650724535692, −9.688544064474393, −8.823526635085885, −8.370023618927428, −7.909071130584404, −7.815888638699164, −6.460534121422247, −6.312902925016955, −5.366486609057036, −4.808557343967551, −4.218175674708734, −3.769386447509718, −3.216960718641765, −2.157008198031361, −1.076597886545584, −0.7144410310207735, 0.7144410310207735, 1.076597886545584, 2.157008198031361, 3.216960718641765, 3.769386447509718, 4.218175674708734, 4.808557343967551, 5.366486609057036, 6.312902925016955, 6.460534121422247, 7.815888638699164, 7.909071130584404, 8.370023618927428, 8.823526635085885, 9.688544064474393, 10.04650724535692, 10.69931737977166, 11.38642806180634, 11.70022466657102, 12.41201934984530, 12.61300981261088, 13.67793043359234, 13.87529323571655, 14.19036695123450, 14.90113894035141

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