Properties

Degree $8$
Conductor $1.032\times 10^{12}$
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  + 6·5-s − 18·17-s + 11·25-s − 14·37-s + 14·49-s + 6·53-s + 6·61-s − 18·73-s − 108·85-s − 6·89-s − 18·101-s + 14·109-s − 48·113-s − 121-s − 6·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 28·169-s + 173-s + 179-s + ⋯
L(s)  = 1  + 2.68·5-s − 4.36·17-s + 11/5·25-s − 2.30·37-s + 2·49-s + 0.824·53-s + 0.768·61-s − 2.10·73-s − 11.7·85-s − 0.635·89-s − 1.79·101-s + 1.34·109-s − 4.51·113-s − 0.0909·121-s − 0.536·125-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 + 2.15·169-s + 0.0760·173-s + 0.0747·179-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 3^{8} \cdot 7^{4}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{1008} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 3^{8} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.2746018601\)
\(L(\frac12)\) \(\approx\) \(0.2746018601\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7$C_2$ \( ( 1 - p T^{2} )^{2} \)
good5$C_2^2$ \( ( 1 - 3 T + 8 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \)
11$C_2^3$ \( 1 + T^{2} - 120 T^{4} + p^{2} T^{6} + p^{4} T^{8} \)
13$C_2^2$ \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 + 9 T + 44 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \)
19$C_2^3$ \( 1 - 31 T^{2} + 600 T^{4} - 31 p^{2} T^{6} + p^{4} T^{8} \)
23$C_2^3$ \( 1 + 25 T^{2} + 96 T^{4} + 25 p^{2} T^{6} + p^{4} T^{8} \)
29$C_2$ \( ( 1 + p T^{2} )^{4} \)
31$C_2^3$ \( 1 - 55 T^{2} + 2064 T^{4} - 55 p^{2} T^{6} + p^{4} T^{8} \)
37$C_2^2$ \( ( 1 + 7 T + 12 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2^3$ \( 1 - 31 T^{2} - 1248 T^{4} - 31 p^{2} T^{6} + p^{4} T^{8} \)
53$C_2^2$ \( ( 1 - 3 T - 44 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \)
59$C_2^3$ \( 1 - 55 T^{2} - 456 T^{4} - 55 p^{2} T^{6} + p^{4} T^{8} \)
61$C_2^2$ \( ( 1 - 3 T + 64 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \)
67$C_2^3$ \( 1 + 113 T^{2} + 8280 T^{4} + 113 p^{2} T^{6} + p^{4} T^{8} \)
71$C_2^2$ \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 + 9 T + 100 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \)
79$C_2^3$ \( 1 + 137 T^{2} + 12528 T^{4} + 137 p^{2} T^{6} + p^{4} T^{8} \)
83$C_2$ \( ( 1 + p T^{2} )^{4} \)
89$C_2^2$ \( ( 1 + 3 T + 92 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 - 182 T^{2} + p^{2} T^{4} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−6.89873427578238172091999417391, −6.84304158859718711240011749337, −6.74439535691993880593196688754, −6.47662414254567900730104175892, −6.15918770743544933030043835052, −6.13501858343338108528619396526, −5.72377639599791445657553349901, −5.56603854341519971068724878277, −5.33877641167049481110865710861, −5.31554391104544217797693165917, −4.74605152955831451290131254109, −4.71846521011306307977979935820, −4.27666683240043686262635125860, −4.13955444668826971207997872736, −3.95333522225174300825059342518, −3.64106911829651362794749256939, −3.01945934665557814152385975183, −2.92123545981488831272916035124, −2.42433849975433527769903920059, −2.20974371425420272304315663410, −2.09701923026291125310806339808, −2.02112644427958919218154372903, −1.40949386715488600927214932143, −1.20590323516215075444832225952, −0.10050117155282918327110425986, 0.10050117155282918327110425986, 1.20590323516215075444832225952, 1.40949386715488600927214932143, 2.02112644427958919218154372903, 2.09701923026291125310806339808, 2.20974371425420272304315663410, 2.42433849975433527769903920059, 2.92123545981488831272916035124, 3.01945934665557814152385975183, 3.64106911829651362794749256939, 3.95333522225174300825059342518, 4.13955444668826971207997872736, 4.27666683240043686262635125860, 4.71846521011306307977979935820, 4.74605152955831451290131254109, 5.31554391104544217797693165917, 5.33877641167049481110865710861, 5.56603854341519971068724878277, 5.72377639599791445657553349901, 6.13501858343338108528619396526, 6.15918770743544933030043835052, 6.47662414254567900730104175892, 6.74439535691993880593196688754, 6.84304158859718711240011749337, 6.89873427578238172091999417391

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