Properties

Degree 4
Conductor $ 3^{2} \cdot 11^{2} \cdot 19^{2} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 2

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 6·5-s + 3·9-s + 11-s + 12·15-s − 4·16-s − 8·23-s + 17·25-s − 4·27-s − 12·31-s − 2·33-s − 18·45-s − 18·47-s + 8·48-s + 11·49-s + 20·53-s − 6·55-s − 16·59-s + 16·67-s + 16·69-s − 24·71-s − 34·75-s + 24·80-s + 5·81-s − 12·89-s + 24·93-s − 20·97-s + ⋯
L(s)  = 1  − 1.15·3-s − 2.68·5-s + 9-s + 0.301·11-s + 3.09·15-s − 16-s − 1.66·23-s + 17/5·25-s − 0.769·27-s − 2.15·31-s − 0.348·33-s − 2.68·45-s − 2.62·47-s + 1.15·48-s + 11/7·49-s + 2.74·53-s − 0.809·55-s − 2.08·59-s + 1.95·67-s + 1.92·69-s − 2.84·71-s − 3.92·75-s + 2.68·80-s + 5/9·81-s − 1.27·89-s + 2.48·93-s − 2.03·97-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 393129 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 393129 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(393129\)    =    \(3^{2} \cdot 11^{2} \cdot 19^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{393129} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  2
Selberg data  =  $(4,\ 393129,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;11,\;19\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{3,\;11,\;19\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad3$C_1$ \( ( 1 + T )^{2} \)
11$C_2$ \( 1 - T + p T^{2} \)
19$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good2$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \)
5$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
23$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
31$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
47$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
67$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.114016907984113880616472369357, −7.47848778675296589128891353641, −7.43755268339255182270866775688, −6.95333653112949028642183365593, −6.41173993136661837405396202601, −5.82201250996795631397735521238, −5.30575361876571555675709307141, −4.64831433194343792963963650578, −4.23023339125662356328251569037, −3.81591641511052084615077615666, −3.57553622245677274525627107284, −2.47849673392805399692938216213, −1.44645188395830487846696686994, 0, 0, 1.44645188395830487846696686994, 2.47849673392805399692938216213, 3.57553622245677274525627107284, 3.81591641511052084615077615666, 4.23023339125662356328251569037, 4.64831433194343792963963650578, 5.30575361876571555675709307141, 5.82201250996795631397735521238, 6.41173993136661837405396202601, 6.95333653112949028642183365593, 7.43755268339255182270866775688, 7.47848778675296589128891353641, 8.114016907984113880616472369357

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