Properties

Degree 8
Conductor $ 2^{24} \cdot 3^{12} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 8·13-s + 10·25-s + 16·37-s − 2·49-s + 16·61-s + 20·73-s + 44·97-s + 16·109-s − 38·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 12·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  − 2.21·13-s + 2·25-s + 2.63·37-s − 2/7·49-s + 2.04·61-s + 2.34·73-s + 4.46·97-s + 1.53·109-s − 3.45·121-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 − 0.923·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(8\)
\( N \)  =  \(2^{24} \cdot 3^{12}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{1728} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((8,\ 2^{24} \cdot 3^{12} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)
\(L(1)\)  \(\approx\)  \(2.169909677\)
\(L(\frac12)\)  \(\approx\)  \(2.169909677\)
\(L(\frac{3}{2})\)   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 \( 1 \)
3 \( 1 \)
good5$C_2^2$ \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
7$C_2^2$ \( ( 1 + T^{2} + p^{2} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 + 19 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{4} \)
17$C_2^2$ \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \)
19$C_2$ \( ( 1 - p T^{2} )^{4} \)
23$C_2^2$ \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 - 47 T^{2} + p^{2} T^{4} )^{2} \)
37$C_2$ \( ( 1 - 4 T + p T^{2} )^{4} \)
41$C_2^2$ \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 + 82 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - 101 T^{2} + p^{2} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + 106 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2$ \( ( 1 - 4 T + p T^{2} )^{4} \)
67$C_2^2$ \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2$ \( ( 1 - 5 T + p T^{2} )^{4} \)
79$C_2$ \( ( 1 - 16 T + p T^{2} )^{2}( 1 + 16 T + p T^{2} )^{2} \)
83$C_2^2$ \( ( 1 + 19 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 - 158 T^{2} + p^{2} T^{4} )^{2} \)
97$C_2$ \( ( 1 - 11 T + p T^{2} )^{4} \)
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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

−6.69866474754708933470649130867, −6.38919823661287754775416674235, −6.17047392633193031559096400435, −6.01832541292989651788101354156, −5.99545367709765374992708686670, −5.45624154831014595108225179134, −5.20441920637936385286345674552, −5.03706830281016540122094677830, −4.87356676384399571926684490045, −4.74342490083514527970801202610, −4.74006368677779757328994762920, −4.09474941180624161284134508973, −4.06792729940957682415109316632, −3.73021094408857367259475447845, −3.52744450935604959873829794120, −3.20812438538059804743906551918, −2.89097057270423967486267410496, −2.63601843455134035974488438199, −2.34910766146292448620771132608, −2.30953153934431804660223957894, −2.08617075506423563662371953794, −1.44122967911840906743579244448, −0.972589239802239879566466175296, −0.912784669434568233243060684461, −0.30718620337990413194323516852, 0.30718620337990413194323516852, 0.912784669434568233243060684461, 0.972589239802239879566466175296, 1.44122967911840906743579244448, 2.08617075506423563662371953794, 2.30953153934431804660223957894, 2.34910766146292448620771132608, 2.63601843455134035974488438199, 2.89097057270423967486267410496, 3.20812438538059804743906551918, 3.52744450935604959873829794120, 3.73021094408857367259475447845, 4.06792729940957682415109316632, 4.09474941180624161284134508973, 4.74006368677779757328994762920, 4.74342490083514527970801202610, 4.87356676384399571926684490045, 5.03706830281016540122094677830, 5.20441920637936385286345674552, 5.45624154831014595108225179134, 5.99545367709765374992708686670, 6.01832541292989651788101354156, 6.17047392633193031559096400435, 6.38919823661287754775416674235, 6.69866474754708933470649130867

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