Properties

Degree 4
Conductor $ 2^{6} \cdot 3^{2} \cdot 19^{2} $
Sign $-1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 1

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 2·3-s + 2·4-s + 4·6-s + 3·9-s + 2·11-s − 4·12-s − 4·16-s − 2·17-s − 6·18-s − 2·19-s − 4·22-s − 25-s − 4·27-s + 8·32-s − 4·33-s + 4·34-s + 6·36-s + 4·38-s − 2·43-s + 4·44-s + 8·48-s + 11·49-s + 2·50-s + 4·51-s + 8·54-s + 4·57-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.15·3-s + 4-s + 1.63·6-s + 9-s + 0.603·11-s − 1.15·12-s − 16-s − 0.485·17-s − 1.41·18-s − 0.458·19-s − 0.852·22-s − 1/5·25-s − 0.769·27-s + 1.41·32-s − 0.696·33-s + 0.685·34-s + 36-s + 0.648·38-s − 0.304·43-s + 0.603·44-s + 1.15·48-s + 11/7·49-s + 0.282·50-s + 0.560·51-s + 1.08·54-s + 0.529·57-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 207936 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 207936 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(207936\)    =    \(2^{6} \cdot 3^{2} \cdot 19^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{207936} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  no
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 207936,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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$C_1$ \( ( 1 + T )^{2} \)
good5$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
7$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
11$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
37$C_2$ \( ( 1 + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
67$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
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

−8.836427381447004052087018681826, −8.562174106312293436599005698498, −7.80605317251250760198226171715, −7.47848778675296589128891353641, −6.97556751348491276200701022616, −6.41173993136661837405396202601, −6.21053798963199366539375101013, −5.43607631650156514852377977246, −4.88619376922667033041566401202, −4.27866732192520382052996066610, −3.81591641511052084615077615666, −2.68139800535164168175122974103, −1.84871910742097553432780734662, −1.10084134564969783818306772778, 0, 1.10084134564969783818306772778, 1.84871910742097553432780734662, 2.68139800535164168175122974103, 3.81591641511052084615077615666, 4.27866732192520382052996066610, 4.88619376922667033041566401202, 5.43607631650156514852377977246, 6.21053798963199366539375101013, 6.41173993136661837405396202601, 6.97556751348491276200701022616, 7.47848778675296589128891353641, 7.80605317251250760198226171715, 8.562174106312293436599005698498, 8.836427381447004052087018681826

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