Properties

Label 8-648e4-1.1-c3e4-0-7
Degree $8$
Conductor $176319369216$
Sign $1$
Analytic cond. $2.13680\times 10^{6}$
Root an. cond. $6.18330$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·5-s + 6·7-s − 10·11-s + 40·13-s − 68·17-s + 68·19-s + 98·23-s + 125·25-s + 120·29-s + 200·31-s − 24·35-s + 960·37-s + 192·41-s + 334·43-s + 300·47-s + 566·49-s − 136·53-s + 40·55-s + 620·59-s − 200·61-s − 160·65-s + 1.40e3·67-s − 2.78e3·71-s + 3.60e3·73-s − 60·77-s + 334·79-s − 500·83-s + ⋯
L(s)  = 1  − 0.357·5-s + 0.323·7-s − 0.274·11-s + 0.853·13-s − 0.970·17-s + 0.821·19-s + 0.888·23-s + 25-s + 0.768·29-s + 1.15·31-s − 0.115·35-s + 4.26·37-s + 0.731·41-s + 1.18·43-s + 0.931·47-s + 1.65·49-s − 0.352·53-s + 0.0980·55-s + 1.36·59-s − 0.419·61-s − 0.305·65-s + 2.56·67-s − 4.64·71-s + 5.77·73-s − 0.0888·77-s + 0.475·79-s − 0.661·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{12} \cdot 3^{16}\)
Sign: $1$
Analytic conductor: \(2.13680\times 10^{6}\)
Root analytic conductor: \(6.18330\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{648} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{12} \cdot 3^{16} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(12.06919650\)
\(L(\frac12)\) \(\approx\) \(12.06919650\)
\(L(\frac{5}{2})\) 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 \( 1 \)
good5$D_4\times C_2$ \( 1 + 4 T - 109 T^{2} - 4 p^{3} T^{3} - 16 p^{3} T^{4} - 4 p^{6} T^{5} - 109 p^{6} T^{6} + 4 p^{9} T^{7} + p^{12} T^{8} \)
7$D_4\times C_2$ \( 1 - 6 T - 530 T^{2} + 720 T^{3} + 190359 T^{4} + 720 p^{3} T^{5} - 530 p^{6} T^{6} - 6 p^{9} T^{7} + p^{12} T^{8} \)
11$D_4\times C_2$ \( 1 + 10 T - 1426 T^{2} - 11360 T^{3} + 424015 T^{4} - 11360 p^{3} T^{5} - 1426 p^{6} T^{6} + 10 p^{9} T^{7} + p^{12} T^{8} \)
13$D_4\times C_2$ \( 1 - 40 T + 31 T^{2} + 113000 T^{3} - 5880248 T^{4} + 113000 p^{3} T^{5} + 31 p^{6} T^{6} - 40 p^{9} T^{7} + p^{12} T^{8} \)
17$D_{4}$ \( ( 1 + 2 p T + 8051 T^{2} + 2 p^{4} T^{3} + p^{6} T^{4} )^{2} \)
19$D_{4}$ \( ( 1 - 34 T + 10782 T^{2} - 34 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 - 98 T - 470 p T^{2} + 384160 T^{3} + 151843639 T^{4} + 384160 p^{3} T^{5} - 470 p^{7} T^{6} - 98 p^{9} T^{7} + p^{12} T^{8} \)
29$D_4\times C_2$ \( 1 - 120 T - 22369 T^{2} + 1441080 T^{3} + 405934440 T^{4} + 1441080 p^{3} T^{5} - 22369 p^{6} T^{6} - 120 p^{9} T^{7} + p^{12} T^{8} \)
31$D_4\times C_2$ \( 1 - 200 T - 21326 T^{2} - 348800 T^{3} + 1681734595 T^{4} - 348800 p^{3} T^{5} - 21326 p^{6} T^{6} - 200 p^{9} T^{7} + p^{12} T^{8} \)
37$D_{4}$ \( ( 1 - 480 T + 152585 T^{2} - 480 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
41$D_4\times C_2$ \( 1 - 192 T - 97294 T^{2} + 707328 T^{3} + 10707560979 T^{4} + 707328 p^{3} T^{5} - 97294 p^{6} T^{6} - 192 p^{9} T^{7} + p^{12} T^{8} \)
43$D_4\times C_2$ \( 1 - 334 T + 5278 T^{2} + 17613824 T^{3} - 3895832657 T^{4} + 17613824 p^{3} T^{5} + 5278 p^{6} T^{6} - 334 p^{9} T^{7} + p^{12} T^{8} \)
47$D_4\times C_2$ \( 1 - 300 T + 46130 T^{2} + 49132800 T^{3} - 18198708429 T^{4} + 49132800 p^{3} T^{5} + 46130 p^{6} T^{6} - 300 p^{9} T^{7} + p^{12} T^{8} \)
53$D_{4}$ \( ( 1 + 68 T + 166814 T^{2} + 68 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
59$D_4\times C_2$ \( 1 - 620 T - 117814 T^{2} - 56702720 T^{3} + 131090680555 T^{4} - 56702720 p^{3} T^{5} - 117814 p^{6} T^{6} - 620 p^{9} T^{7} + p^{12} T^{8} \)
61$D_4\times C_2$ \( 1 + 200 T - 265937 T^{2} - 29605000 T^{3} + 32997833608 T^{4} - 29605000 p^{3} T^{5} - 265937 p^{6} T^{6} + 200 p^{9} T^{7} + p^{12} T^{8} \)
67$D_4\times C_2$ \( 1 - 1406 T + 896710 T^{2} - 672911600 T^{3} + 481654667839 T^{4} - 672911600 p^{3} T^{5} + 896710 p^{6} T^{6} - 1406 p^{9} T^{7} + p^{12} T^{8} \)
71$D_{4}$ \( ( 1 + 1390 T + 1197686 T^{2} + 1390 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
73$D_{4}$ \( ( 1 - 1802 T + 1548039 T^{2} - 1802 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
79$D_4\times C_2$ \( 1 - 334 T - 97322 T^{2} + 259584800 T^{3} - 254458895321 T^{4} + 259584800 p^{3} T^{5} - 97322 p^{6} T^{6} - 334 p^{9} T^{7} + p^{12} T^{8} \)
83$D_4\times C_2$ \( 1 + 500 T - 249670 T^{2} - 321952000 T^{3} - 220217014469 T^{4} - 321952000 p^{3} T^{5} - 249670 p^{6} T^{6} + 500 p^{9} T^{7} + p^{12} T^{8} \)
89$D_{4}$ \( ( 1 + 90 T + 1337659 T^{2} + 90 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 - 200 T + 124690 T^{2} + 382007200 T^{3} - 862306528829 T^{4} + 382007200 p^{3} T^{5} + 124690 p^{6} T^{6} - 200 p^{9} T^{7} + p^{12} T^{8} \)
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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

−7.38619296656606172647038009706, −6.88832444732869969507061377588, −6.62770499674834878822460436937, −6.44516247025440567188579085319, −6.39529178170485757368413336940, −5.89557368260111530536584771352, −5.76850723464646742918208016649, −5.49233250671400744812954898541, −5.42937844469274316680780321468, −4.73467184683474604021421761653, −4.63579816145004138540532165432, −4.48377993178721356262317994049, −4.37007991785573504338928551822, −4.06976479400176359727968306524, −3.53230893313971585511597178639, −3.27584649677394159734917413949, −3.21251882227317435614488628154, −2.58670923024984572116775579271, −2.53245855874569179348782902801, −2.12049258378068866453526584073, −2.03060484627319535384120771705, −0.946118877479188781225987376750, −0.918429456703710330615923380638, −0.879049266342801329050244487834, −0.58082202613745626440706377920, 0.58082202613745626440706377920, 0.879049266342801329050244487834, 0.918429456703710330615923380638, 0.946118877479188781225987376750, 2.03060484627319535384120771705, 2.12049258378068866453526584073, 2.53245855874569179348782902801, 2.58670923024984572116775579271, 3.21251882227317435614488628154, 3.27584649677394159734917413949, 3.53230893313971585511597178639, 4.06976479400176359727968306524, 4.37007991785573504338928551822, 4.48377993178721356262317994049, 4.63579816145004138540532165432, 4.73467184683474604021421761653, 5.42937844469274316680780321468, 5.49233250671400744812954898541, 5.76850723464646742918208016649, 5.89557368260111530536584771352, 6.39529178170485757368413336940, 6.44516247025440567188579085319, 6.62770499674834878822460436937, 6.88832444732869969507061377588, 7.38619296656606172647038009706

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