Properties

Label 8-9800e4-1.1-c1e4-0-10
Degree $8$
Conductor $9.224\times 10^{15}$
Sign $1$
Analytic cond. $3.74983\times 10^{7}$
Root an. cond. $8.84609$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·9-s − 4·11-s + 4·23-s − 8·29-s − 8·37-s + 4·53-s + 12·67-s − 12·79-s + 14·81-s + 24·99-s − 28·109-s − 16·113-s − 34·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 46·169-s + 173-s + 179-s + 181-s + 191-s + ⋯
L(s)  = 1  − 2·9-s − 1.20·11-s + 0.834·23-s − 1.48·29-s − 1.31·37-s + 0.549·53-s + 1.46·67-s − 1.35·79-s + 14/9·81-s + 2.41·99-s − 2.68·109-s − 1.50·113-s − 3.09·121-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 − 3.53·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{12} \cdot 5^{8} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(3.74983\times 10^{7}\)
Root analytic conductor: \(8.84609\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{12} \cdot 5^{8} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good3$D_4\times C_2$ \( 1 + 2 p T^{2} + 22 T^{4} + 2 p^{3} T^{6} + p^{4} T^{8} \)
11$C_2$ \( ( 1 + T + p T^{2} )^{4} \)
13$D_4\times C_2$ \( 1 + 46 T^{2} + 862 T^{4} + 46 p^{2} T^{6} + p^{4} T^{8} \)
17$D_4\times C_2$ \( 1 + 44 T^{2} + 982 T^{4} + 44 p^{2} T^{6} + p^{4} T^{8} \)
19$C_2^2$ \( ( 1 + 28 T^{2} + p^{2} T^{4} )^{2} \)
23$C_2$ \( ( 1 - T + p T^{2} )^{4} \)
29$D_{4}$ \( ( 1 + 4 T + 17 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
31$D_4\times C_2$ \( 1 + 70 T^{2} + 2742 T^{4} + 70 p^{2} T^{6} + p^{4} T^{8} \)
37$D_{4}$ \( ( 1 + 4 T + 73 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
41$D_4\times C_2$ \( 1 + 70 T^{2} + 2382 T^{4} + 70 p^{2} T^{6} + p^{4} T^{8} \)
43$C_2^2$ \( ( 1 + 81 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 + 92 T^{2} + p^{2} T^{4} )^{2} \)
53$D_{4}$ \( ( 1 - 2 T + 102 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
59$D_4\times C_2$ \( 1 + 110 T^{2} + 6342 T^{4} + 110 p^{2} T^{6} + p^{4} T^{8} \)
61$D_4\times C_2$ \( 1 + 214 T^{2} + 18766 T^{4} + 214 p^{2} T^{6} + p^{4} T^{8} \)
67$D_{4}$ \( ( 1 - 6 T + 123 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 + 97 T^{2} + p^{2} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 + 118 T^{2} + 8014 T^{4} + 118 p^{2} T^{6} + p^{4} T^{8} \)
79$D_{4}$ \( ( 1 + 6 T + 87 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 + 262 T^{2} + 29814 T^{4} + 262 p^{2} T^{6} + p^{4} T^{8} \)
89$D_4\times C_2$ \( 1 + 326 T^{2} + 42286 T^{4} + 326 p^{2} T^{6} + p^{4} T^{8} \)
97$D_4\times C_2$ \( 1 + 214 T^{2} + 24142 T^{4} + 214 p^{2} T^{6} + p^{4} T^{8} \)
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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

−5.82027171619566868292604810163, −5.30047778282929059809820635060, −5.28575587748807214741612295194, −5.26805504511782616873915266734, −5.23835750415948705060307146039, −4.94052976760129333548520506552, −4.80498803813865549819131705363, −4.50259622043833041673650467347, −4.24261995778889080813801947130, −3.85275561895575502020963355447, −3.81018121098235062735919730276, −3.75319665529572588592484055812, −3.74857078769419985469867525394, −3.16512698255567273629083516220, −3.00391618008024732161545111212, −2.90453125945152679497049463070, −2.85038158540669465438109503658, −2.37429289353442788701379557178, −2.35987235323603329676471453417, −2.24933264655826249499051095713, −1.98776374345381751251063445401, −1.51671529519761667543628641518, −1.27502709884268367960738509428, −1.05758870008561662177969168979, −1.00270290093464359016966584370, 0, 0, 0, 0, 1.00270290093464359016966584370, 1.05758870008561662177969168979, 1.27502709884268367960738509428, 1.51671529519761667543628641518, 1.98776374345381751251063445401, 2.24933264655826249499051095713, 2.35987235323603329676471453417, 2.37429289353442788701379557178, 2.85038158540669465438109503658, 2.90453125945152679497049463070, 3.00391618008024732161545111212, 3.16512698255567273629083516220, 3.74857078769419985469867525394, 3.75319665529572588592484055812, 3.81018121098235062735919730276, 3.85275561895575502020963355447, 4.24261995778889080813801947130, 4.50259622043833041673650467347, 4.80498803813865549819131705363, 4.94052976760129333548520506552, 5.23835750415948705060307146039, 5.26805504511782616873915266734, 5.28575587748807214741612295194, 5.30047778282929059809820635060, 5.82027171619566868292604810163

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