Properties

Degree 8
Conductor $ 2^{12} \cdot 3^{4} $
Sign $1$
Motivic weight 7
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 36·3-s − 160·4-s − 3.40e3·9-s + 5.76e3·12-s + 9.21e3·16-s + 4.62e4·19-s − 2.93e5·25-s + 2.12e5·27-s + 5.44e5·36-s + 1.98e6·43-s − 3.31e5·48-s − 1.50e6·49-s − 1.66e6·57-s + 1.14e6·64-s + 5.60e6·67-s − 8.89e6·73-s + 1.05e7·75-s − 7.40e6·76-s + 6.37e6·81-s + 2.74e7·97-s + 4.69e7·100-s − 3.40e7·108-s + 7.15e7·121-s + 127-s − 7.12e7·129-s + 131-s + 137-s + ⋯
L(s)  = 1  − 0.769·3-s − 5/4·4-s − 1.55·9-s + 0.962·12-s + 9/16·16-s + 1.54·19-s − 3.75·25-s + 2.08·27-s + 1.94·36-s + 3.79·43-s − 0.433·48-s − 1.82·49-s − 1.19·57-s + 0.546·64-s + 2.27·67-s − 2.67·73-s + 2.89·75-s − 1.93·76-s + 1.33·81-s + 3.05·97-s + 4.69·100-s − 2.60·108-s + 3.67·121-s − 2.92·129-s − 7/8·144-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(8\)
\( N \)  =  \(331776\)    =    \(2^{12} \cdot 3^{4}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(7\)
character  :  induced by $\chi_{24} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(8,\ 331776,\ (\ :7/2, 7/2, 7/2, 7/2),\ 1)$
$L(4)$  $\approx$  $0.957440$
$L(\frac12)$  $\approx$  $0.957440$
$L(\frac{9}{2})$   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$C_2^2$ \( 1 + 5 p^{5} T^{2} + p^{14} T^{4} \)
3$C_2$ \( ( 1 + 2 p^{2} T + p^{7} T^{2} )^{2} \)
good5$C_2^2$ \( ( 1 + 5866 p^{2} T^{2} + p^{14} T^{4} )^{2} \)
7$C_2^2$ \( ( 1 + 751570 T^{2} + p^{14} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - 35789342 T^{2} + p^{14} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - 1741490 p T^{2} + p^{14} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 - 61393730 T^{2} + p^{14} T^{4} )^{2} \)
19$C_2$ \( ( 1 - 11570 T + p^{7} T^{2} )^{4} \)
23$C_2^2$ \( ( 1 + 3725533390 T^{2} + p^{14} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 + 31499935018 T^{2} + p^{14} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 - 49885402622 T^{2} + p^{14} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 - 80259050 p^{2} T^{2} + p^{14} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 - 226584602162 T^{2} + p^{14} T^{4} )^{2} \)
43$C_2$ \( ( 1 - 495062 T + p^{7} T^{2} )^{4} \)
47$C_2^2$ \( ( 1 - 277104744290 T^{2} + p^{14} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 + 2021761791610 T^{2} + p^{14} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 - 2902252328702 T^{2} + p^{14} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 - 3094549289642 T^{2} + p^{14} T^{4} )^{2} \)
67$C_2$ \( ( 1 - 1400126 T + p^{7} T^{2} )^{4} \)
71$C_2^2$ \( ( 1 + 5449089705838 T^{2} + p^{14} T^{4} )^{2} \)
73$C_2$ \( ( 1 + 2223598 T + p^{7} T^{2} )^{4} \)
79$C_2^2$ \( ( 1 - 7153320805022 T^{2} + p^{14} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 - 45191963757710 T^{2} + p^{14} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 - 54294758858162 T^{2} + p^{14} T^{4} )^{2} \)
97$C_2$ \( ( 1 - 6867926 T + p^{7} T^{2} )^{4} \)
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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

−11.65137612415817674435213595086, −11.42075132100137735024749145402, −11.31614524529457352486230619775, −10.63083006754214581736471169211, −10.34253788759327420059655150819, −9.778695930279828123962991679982, −9.494858364926015074978772945243, −9.276440927987274633659092311121, −8.918375286249469763475252312775, −8.251205567555538711877387299895, −8.143439394125973515482947900445, −7.58121902724244265860583281121, −7.35151700441152107895653609277, −6.50417438138518906960741438036, −5.87852117642755702074583407047, −5.77426551844920887517133284403, −5.54449617137690454772694594239, −4.83073572100013523590809034938, −4.41130201036643473029754139859, −3.78010783986117100231522831858, −3.31882280413394818242892203812, −2.56786104934561066993204887814, −1.83175964532965028086829800452, −0.59242553597870120860012054756, −0.49567757561840387007010790067, 0.49567757561840387007010790067, 0.59242553597870120860012054756, 1.83175964532965028086829800452, 2.56786104934561066993204887814, 3.31882280413394818242892203812, 3.78010783986117100231522831858, 4.41130201036643473029754139859, 4.83073572100013523590809034938, 5.54449617137690454772694594239, 5.77426551844920887517133284403, 5.87852117642755702074583407047, 6.50417438138518906960741438036, 7.35151700441152107895653609277, 7.58121902724244265860583281121, 8.143439394125973515482947900445, 8.251205567555538711877387299895, 8.918375286249469763475252312775, 9.276440927987274633659092311121, 9.494858364926015074978772945243, 9.778695930279828123962991679982, 10.34253788759327420059655150819, 10.63083006754214581736471169211, 11.31614524529457352486230619775, 11.42075132100137735024749145402, 11.65137612415817674435213595086

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