Properties

Degree 4
Conductor $ 3^{3} \cdot 7^{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  + 3-s − 3·4-s − 4·5-s + 9-s − 3·12-s − 4·15-s + 5·16-s − 12·17-s + 12·20-s + 2·25-s + 27-s − 3·36-s − 20·37-s − 4·41-s − 8·43-s − 4·45-s + 24·47-s + 5·48-s − 7·49-s − 12·51-s − 24·59-s + 12·60-s − 3·64-s − 8·67-s + 36·68-s + 2·75-s − 20·80-s + ⋯
L(s)  = 1  + 0.577·3-s − 3/2·4-s − 1.78·5-s + 1/3·9-s − 0.866·12-s − 1.03·15-s + 5/4·16-s − 2.91·17-s + 2.68·20-s + 2/5·25-s + 0.192·27-s − 1/2·36-s − 3.28·37-s − 0.624·41-s − 1.21·43-s − 0.596·45-s + 3.50·47-s + 0.721·48-s − 49-s − 1.68·51-s − 3.12·59-s + 1.54·60-s − 3/8·64-s − 0.977·67-s + 4.36·68-s + 0.230·75-s − 2.23·80-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 477603 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 477603 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(477603\)    =    \(3^{3} \cdot 7^{2} \cdot 19^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{477603} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  2
Selberg data  =  $(4,\ 477603,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

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

Imaginary part of the first few zeros on the critical line

−8.263513432862467471567669978548, −7.75382334304440982742558846703, −7.34752150402139467995584348966, −6.78418502492955034999702547464, −6.46787589567348650226556085189, −5.56436473344957075485364042125, −4.94046317313085415521526307990, −4.56991321080597356703354311172, −4.20789833671809510725213788268, −3.70760760284754853820305649861, −3.44365518362789223021999993571, −2.47817878081925227300773446057, −1.65980873553023464631625543363, 0, 0, 1.65980873553023464631625543363, 2.47817878081925227300773446057, 3.44365518362789223021999993571, 3.70760760284754853820305649861, 4.20789833671809510725213788268, 4.56991321080597356703354311172, 4.94046317313085415521526307990, 5.56436473344957075485364042125, 6.46787589567348650226556085189, 6.78418502492955034999702547464, 7.34752150402139467995584348966, 7.75382334304440982742558846703, 8.263513432862467471567669978548

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