Properties

Degree 8
Conductor $ 2^{12} \cdot 3^{4} $
Sign $1$
Motivic weight 2
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·4-s + 16·7-s − 10·9-s + 20·16-s − 36·25-s − 96·28-s − 16·31-s + 60·36-s − 36·49-s − 160·63-s − 24·64-s − 24·73-s + 496·79-s + 19·81-s + 472·97-s + 216·100-s − 432·103-s + 320·112-s − 340·121-s + 96·124-s + 127-s + 131-s + 137-s + 139-s − 200·144-s + 149-s + 151-s + ⋯
L(s)  = 1  − 3/2·4-s + 16/7·7-s − 1.11·9-s + 5/4·16-s − 1.43·25-s − 3.42·28-s − 0.516·31-s + 5/3·36-s − 0.734·49-s − 2.53·63-s − 3/8·64-s − 0.328·73-s + 6.27·79-s + 0.234·81-s + 4.86·97-s + 2.15·100-s − 4.19·103-s + 20/7·112-s − 2.80·121-s + 0.774·124-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 1.38·144-s + 0.00671·149-s + 0.00662·151-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(8\)
\( N \)  =  \(331776\)    =    \(2^{12} \cdot 3^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(2\)
character  :  induced by $\chi_{24} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((8,\ 331776,\ (\ :1, 1, 1, 1),\ 1)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.600764\)
\(L(\frac12)\)  \(\approx\)  \(0.600764\)
\(L(2)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3\}$,\(F_p(T)\) is a polynomial of degree 8. If $p \in \{2,\;3\}$, then $F_p(T)$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_2^2$ \( 1 + 3 p T^{2} + p^{4} T^{4} \)
3$C_2^2$ \( 1 + 10 T^{2} + p^{4} T^{4} \)
good5$C_2^2$ \( ( 1 + 18 T^{2} + p^{4} T^{4} )^{2} \)
7$C_2$ \( ( 1 - 4 T + p^{2} T^{2} )^{4} \)
11$C_2^2$ \( ( 1 + 170 T^{2} + p^{4} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - 226 T^{2} + p^{4} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 - 354 T^{2} + p^{4} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 - 694 T^{2} + p^{4} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 - 162 T^{2} + p^{4} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 + 1394 T^{2} + p^{4} T^{4} )^{2} \)
31$C_2$ \( ( 1 + 4 T + p^{2} T^{2} )^{4} \)
37$C_2^2$ \( ( 1 + 62 T^{2} + p^{4} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 - 2466 T^{2} + p^{4} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 - 3670 T^{2} + p^{4} T^{4} )^{2} \)
47$C_1$$\times$$C_1$ \( ( 1 - p T )^{4}( 1 + p T )^{4} \)
53$C_2^2$ \( ( 1 + 3026 T^{2} + p^{4} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + 4650 T^{2} + p^{4} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 + 1630 T^{2} + p^{4} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 - 6710 T^{2} + p^{4} T^{4} )^{2} \)
71$C_2$ \( ( 1 - 110 T + p^{2} T^{2} )^{2}( 1 + 110 T + p^{2} T^{2} )^{2} \)
73$C_2$ \( ( 1 + 6 T + p^{2} T^{2} )^{4} \)
79$C_2$ \( ( 1 - 124 T + p^{2} T^{2} )^{4} \)
83$C_2^2$ \( ( 1 + 13770 T^{2} + p^{4} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 - 4866 T^{2} + p^{4} T^{4} )^{2} \)
97$C_2$ \( ( 1 - 118 T + p^{2} T^{2} )^{4} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−13.30585491727059322223027458989, −13.06787330122284560108302223006, −12.65300169127315327592621779684, −11.94041167474267908976916677818, −11.89746978123167618675420931678, −11.67471531999220073326774440554, −11.04438544989209094837587220671, −10.79036843865912096372979466727, −10.72062752973857449709958500144, −9.846670408141429861209173533844, −9.544275704799949303938687941939, −9.292175672440491200167963865656, −8.798698392185051621009315897507, −8.351408340978118145097745896039, −8.122046395595495303490504147548, −7.82115244114720862878575879135, −7.55293857840674426208737803386, −6.52670727400072296712392964723, −6.14447145950804349033210209199, −5.38400542037203543276090817067, −4.98220427408142433639761568124, −4.89793294545700987547852631759, −4.05866501531734767507822894176, −3.45864190635624545873021523023, −1.99519762672624888677905786603, 1.99519762672624888677905786603, 3.45864190635624545873021523023, 4.05866501531734767507822894176, 4.89793294545700987547852631759, 4.98220427408142433639761568124, 5.38400542037203543276090817067, 6.14447145950804349033210209199, 6.52670727400072296712392964723, 7.55293857840674426208737803386, 7.82115244114720862878575879135, 8.122046395595495303490504147548, 8.351408340978118145097745896039, 8.798698392185051621009315897507, 9.292175672440491200167963865656, 9.544275704799949303938687941939, 9.846670408141429861209173533844, 10.72062752973857449709958500144, 10.79036843865912096372979466727, 11.04438544989209094837587220671, 11.67471531999220073326774440554, 11.89746978123167618675420931678, 11.94041167474267908976916677818, 12.65300169127315327592621779684, 13.06787330122284560108302223006, 13.30585491727059322223027458989

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