Properties

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

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·4-s − 4·13-s + 4·25-s − 4·37-s + 22·49-s − 8·52-s + 44·61-s − 8·64-s − 4·73-s − 52·97-s + 8·100-s − 40·109-s + 4·121-s + 127-s + 131-s + 137-s + 139-s − 8·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 42·169-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  + 4-s − 1.10·13-s + 4/5·25-s − 0.657·37-s + 22/7·49-s − 1.10·52-s + 5.63·61-s − 64-s − 0.468·73-s − 5.27·97-s + 4/5·100-s − 3.83·109-s + 4/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.657·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 3.23·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(8\)
\( N \)  =  \(2^{8} \cdot 3^{12}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{108} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(8,\ 2^{8} \cdot 3^{12} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$
$L(1)$  $\approx$  $1.21164$
$L(\frac12)$  $\approx$  $1.21164$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3\}$,\(F_p(T)\) 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$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
3 \( 1 \)
good5$C_2^2$ \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \)
7$C_2$ \( ( 1 - 5 T + p T^{2} )^{2}( 1 + 5 T + p T^{2} )^{2} \)
11$C_2^2$ \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2$ \( ( 1 + T + p T^{2} )^{4} \)
17$C_2^2$ \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \)
19$C_2$ \( ( 1 - 7 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \)
23$C_2^2$ \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \)
37$C_2$ \( ( 1 + T + p T^{2} )^{4} \)
41$C_2^2$ \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + 94 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2$ \( ( 1 - 11 T + p T^{2} )^{4} \)
67$C_2$ \( ( 1 - 11 T + p T^{2} )^{2}( 1 + 11 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{4} \)
73$C_2$ \( ( 1 + T + p T^{2} )^{4} \)
79$C_2^2$ \( ( 1 - 155 T^{2} + p^{2} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 - 170 T^{2} + p^{2} T^{4} )^{2} \)
97$C_2$ \( ( 1 + 13 T + p T^{2} )^{4} \)
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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

−10.07056945536151663804908059052, −10.03972817666480461842307116961, −9.579514382579032216601020781947, −9.231739868664361363571273147442, −9.163159153657917161863809723513, −8.581188624764473361244357411680, −8.285813732700835825090870302135, −8.281721265933742721789710956170, −7.80511589406951707490348315572, −7.16132092439863928675358722446, −7.10773983113878985981094762058, −7.03252381209453066427768669185, −6.73714895808816940727636904501, −6.29241448203311121119305129276, −5.77710144178911363071619859598, −5.46966003214090417135767742386, −5.28584919165466854143316822829, −4.90535018477208437957651904602, −4.31239716399917921215721847652, −3.83954399237016636898470666470, −3.78108429902194902533382754795, −2.77828702923805117006288224857, −2.53358746501207533779415162693, −2.31677886908070896022890697102, −1.28738494146991810369837546020, 1.28738494146991810369837546020, 2.31677886908070896022890697102, 2.53358746501207533779415162693, 2.77828702923805117006288224857, 3.78108429902194902533382754795, 3.83954399237016636898470666470, 4.31239716399917921215721847652, 4.90535018477208437957651904602, 5.28584919165466854143316822829, 5.46966003214090417135767742386, 5.77710144178911363071619859598, 6.29241448203311121119305129276, 6.73714895808816940727636904501, 7.03252381209453066427768669185, 7.10773983113878985981094762058, 7.16132092439863928675358722446, 7.80511589406951707490348315572, 8.281721265933742721789710956170, 8.285813732700835825090870302135, 8.581188624764473361244357411680, 9.163159153657917161863809723513, 9.231739868664361363571273147442, 9.579514382579032216601020781947, 10.03972817666480461842307116961, 10.07056945536151663804908059052

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