Properties

Degree 8
Conductor $ 3^{4} \cdot 7^{4} $
Sign $1$
Motivic weight 3
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 + 28·7-s + 48·9-s − 125·16-s − 296·25-s − 56·28-s − 96·36-s + 920·37-s + 176·43-s − 98·49-s + 1.34e3·63-s + 380·64-s − 256·67-s − 1.76e3·79-s + 1.57e3·81-s + 592·100-s − 7.48e3·109-s − 3.50e3·112-s + 3.14e3·121-s + 127-s + 131-s + 137-s + 139-s − 6.00e3·144-s − 1.84e3·148-s + 149-s + 151-s + ⋯
L(s)  = 1  − 1/4·4-s + 1.51·7-s + 16/9·9-s − 1.95·16-s − 2.36·25-s − 0.377·28-s − 4/9·36-s + 4.08·37-s + 0.624·43-s − 2/7·49-s + 2.68·63-s + 0.742·64-s − 0.466·67-s − 2.51·79-s + 2.16·81-s + 0.591·100-s − 6.57·109-s − 2.95·112-s + 2.36·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s − 3.47·144-s − 1.02·148-s + 0.000549·149-s + 0.000538·151-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(8\)
\( N \)  =  \(194481\)    =    \(3^{4} \cdot 7^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(3\)
character  :  induced by $\chi_{21} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(8,\ 194481,\ (\ :3/2, 3/2, 3/2, 3/2),\ 1)$
$L(2)$  $\approx$  $1.46418$
$L(\frac12)$  $\approx$  $1.46418$
$L(\frac{5}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;7\}$, \(F_p\) is a polynomial of degree 8. If $p \in \{3,\;7\}$, then $F_p$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p$
bad3$C_2^2$ \( 1 - 16 p T^{2} + p^{6} T^{4} \)
7$C_2$ \( ( 1 - 2 p T + p^{3} T^{2} )^{2} \)
good2$C_2^2$ \( ( 1 + T^{2} + p^{6} T^{4} )^{2} \)
5$C_2^2$ \( ( 1 + 148 T^{2} + p^{6} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - 1574 T^{2} + p^{6} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - 1220 T^{2} + p^{6} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 + 362 p T^{2} + p^{6} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 - 12368 T^{2} + p^{6} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 - 16106 T^{2} + p^{6} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 45446 T^{2} + p^{6} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 + 5314 T^{2} + p^{6} T^{4} )^{2} \)
37$C_2$ \( ( 1 - 230 T + p^{3} T^{2} )^{4} \)
41$C_2^2$ \( ( 1 + 117850 T^{2} + p^{6} T^{4} )^{2} \)
43$C_2$ \( ( 1 - 44 T + p^{3} T^{2} )^{4} \)
47$C_2^2$ \( ( 1 + 89734 T^{2} + p^{6} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - 255254 T^{2} + p^{6} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + 393520 T^{2} + p^{6} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 - 448916 T^{2} + p^{6} T^{4} )^{2} \)
67$C_2$ \( ( 1 + 64 T + p^{3} T^{2} )^{4} \)
71$C_2^2$ \( ( 1 - 502574 T^{2} + p^{6} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 - 770258 T^{2} + p^{6} T^{4} )^{2} \)
79$C_2$ \( ( 1 + 442 T + p^{3} T^{2} )^{4} \)
83$C_2^2$ \( ( 1 + 898672 T^{2} + p^{6} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 + 1174930 T^{2} + p^{6} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 - 631850 T^{2} + p^{6} T^{4} )^{2} \)
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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

−13.21969591693071097044109254109, −13.04970478817152439150133503826, −12.92106877133152773390542059963, −12.10185213718704110317220700395, −11.79678700743563857217526952473, −11.48775398808851655567942659096, −11.34292367802797560617698743003, −10.71122465500148351413525987822, −10.60292365581292974983140671336, −9.727412047358681193722752408225, −9.711212012730355163794789098556, −9.324080705887293607891889333454, −8.956709544708354989406186761080, −7.995433783463858024492288216964, −7.912000685252542662375479189858, −7.83455932287817335780675760359, −6.85989490220117982685325390528, −6.82973460345722134832736390116, −5.96506418283386290183573186618, −5.49719705219397201986108773057, −4.59500727037303649384588985717, −4.25110069166822848048650606589, −4.21846996750222375127956771415, −2.50041469302151453672513236121, −1.58894507498030896984846429630, 1.58894507498030896984846429630, 2.50041469302151453672513236121, 4.21846996750222375127956771415, 4.25110069166822848048650606589, 4.59500727037303649384588985717, 5.49719705219397201986108773057, 5.96506418283386290183573186618, 6.82973460345722134832736390116, 6.85989490220117982685325390528, 7.83455932287817335780675760359, 7.912000685252542662375479189858, 7.995433783463858024492288216964, 8.956709544708354989406186761080, 9.324080705887293607891889333454, 9.711212012730355163794789098556, 9.727412047358681193722752408225, 10.60292365581292974983140671336, 10.71122465500148351413525987822, 11.34292367802797560617698743003, 11.48775398808851655567942659096, 11.79678700743563857217526952473, 12.10185213718704110317220700395, 12.92106877133152773390542059963, 13.04970478817152439150133503826, 13.21969591693071097044109254109

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