Properties

Degree 8
Conductor $ 2^{12} \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  + 3·4-s + 4·7-s + 5·16-s + 18·25-s + 12·28-s − 28·31-s − 18·49-s + 3·64-s + 12·73-s + 16·79-s + 28·97-s + 54·100-s + 20·112-s + 26·121-s − 84·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4·169-s + 173-s + 72·175-s + ⋯
L(s)  = 1  + 3/2·4-s + 1.51·7-s + 5/4·16-s + 18/5·25-s + 2.26·28-s − 5.02·31-s − 2.57·49-s + 3/8·64-s + 1.40·73-s + 1.80·79-s + 2.84·97-s + 27/5·100-s + 1.88·112-s + 2.36·121-s − 7.54·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.307·169-s + 0.0760·173-s + 5.44·175-s + ⋯

Functional equation

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

Invariants

\( d \)  =  \(8\)
\( N \)  =  \(2^{12} \cdot 3^{12}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{216} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(8,\ 2^{12} \cdot 3^{12} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$
$L(1)$  $\approx$  $2.95136$
$L(\frac12)$  $\approx$  $2.95136$
$L(\frac{3}{2})$   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$C_2^2$ \( 1 - 3 T^{2} + p^{2} T^{4} \)
3 \( 1 \)
good5$C_2^2$ \( ( 1 - 9 T^{2} + p^{2} T^{4} )^{2} \)
7$C_2$ \( ( 1 - T + p T^{2} )^{4} \)
11$C_2^2$ \( ( 1 - 13 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 + 18 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2$ \( ( 1 + 7 T + p T^{2} )^{4} \)
37$C_2^2$ \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 + 54 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{4} \)
53$C_2^2$ \( ( 1 - 25 T^{2} + p^{2} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 - 102 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2$ \( ( 1 - p T^{2} )^{4} \)
67$C_2$ \( ( 1 - p T^{2} )^{4} \)
71$C_2^2$ \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2$ \( ( 1 - 3 T + p T^{2} )^{4} \)
79$C_2$ \( ( 1 - 4 T + p T^{2} )^{4} \)
83$C_2^2$ \( ( 1 - 117 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 + 66 T^{2} + p^{2} T^{4} )^{2} \)
97$C_2$ \( ( 1 - 7 T + p T^{2} )^{4} \)
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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

−9.172258595227955719917877285612, −8.671047688937859713956696401459, −8.384929088395324122507493186650, −8.166995065736436581086886298015, −8.034773514306085629413047787340, −7.61564852060630932565362852007, −7.23618476599573020784224113199, −7.08058733960919291030355702453, −6.98990989642052134622472713068, −6.83491221339109599861053635000, −6.16423515543451171126966433786, −6.00240492982699520370941990099, −5.84368210366236991635008430520, −5.26727410910163239632552261105, −4.91339111600050188067292608053, −4.88329052261943764743593500737, −4.74187021437915437324284521934, −4.00212274472776714422021322012, −3.46551182701967147505360841496, −3.27823863425210543679099280084, −3.15764758579145291204500908366, −2.24793440843523719707062138465, −2.02578688338605839856043544083, −1.75430672134053610056750135567, −1.09597606645478970806256319479, 1.09597606645478970806256319479, 1.75430672134053610056750135567, 2.02578688338605839856043544083, 2.24793440843523719707062138465, 3.15764758579145291204500908366, 3.27823863425210543679099280084, 3.46551182701967147505360841496, 4.00212274472776714422021322012, 4.74187021437915437324284521934, 4.88329052261943764743593500737, 4.91339111600050188067292608053, 5.26727410910163239632552261105, 5.84368210366236991635008430520, 6.00240492982699520370941990099, 6.16423515543451171126966433786, 6.83491221339109599861053635000, 6.98990989642052134622472713068, 7.08058733960919291030355702453, 7.23618476599573020784224113199, 7.61564852060630932565362852007, 8.034773514306085629413047787340, 8.166995065736436581086886298015, 8.384929088395324122507493186650, 8.671047688937859713956696401459, 9.172258595227955719917877285612

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