Properties

Degree 4
Conductor $ 2^{4} \cdot 3^{2} \cdot 19^{3} $
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  − 2·2-s − 2·3-s + 2·4-s − 6·5-s + 4·6-s + 3·9-s + 12·10-s − 4·12-s + 12·15-s − 4·16-s − 2·17-s − 6·18-s − 19-s − 12·20-s + 17·25-s − 4·27-s − 24·30-s − 12·31-s + 8·32-s + 4·34-s + 6·36-s + 2·38-s − 18·45-s + 8·48-s + 11·49-s − 34·50-s + 4·51-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.15·3-s + 4-s − 2.68·5-s + 1.63·6-s + 9-s + 3.79·10-s − 1.15·12-s + 3.09·15-s − 16-s − 0.485·17-s − 1.41·18-s − 0.229·19-s − 2.68·20-s + 17/5·25-s − 0.769·27-s − 4.38·30-s − 2.15·31-s + 1.41·32-s + 0.685·34-s + 36-s + 0.324·38-s − 2.68·45-s + 1.15·48-s + 11/7·49-s − 4.80·50-s + 0.560·51-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 987696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 987696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(987696\)    =    \(2^{4} \cdot 3^{2} \cdot 19^{3}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{987696} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  2
Selberg data  =  $(4,\ 987696,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;19\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;19\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_2$ \( 1 + p T + p T^{2} \)
3$C_1$ \( ( 1 + T )^{2} \)
19$C_1$ \( 1 + T \)
good5$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
11$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
59$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−7.48919464134144900735099782123, −7.47848778675296589128891353641, −7.27079151643991129277821474463, −6.41173993136661837405396202601, −6.25499216179811745555302525608, −5.36600607918508759890630649285, −4.89390739159963849894021052339, −4.42808617434113254239775941766, −3.81591641511052084615077615666, −3.79824334094358549621418265757, −2.80250883164613859029957691965, −1.86642316096709822967327164489, −1.02602590121900935054247645611, 0, 0, 1.02602590121900935054247645611, 1.86642316096709822967327164489, 2.80250883164613859029957691965, 3.79824334094358549621418265757, 3.81591641511052084615077615666, 4.42808617434113254239775941766, 4.89390739159963849894021052339, 5.36600607918508759890630649285, 6.25499216179811745555302525608, 6.41173993136661837405396202601, 7.27079151643991129277821474463, 7.47848778675296589128891353641, 7.48919464134144900735099782123

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