Properties

Label 8-4864e4-1.1-c1e4-0-3
Degree $8$
Conductor $5.597\times 10^{14}$
Sign $1$
Analytic cond. $2.27553\times 10^{6}$
Root an. cond. $6.23211$
Motivic weight $1$
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  − 12·9-s + 6·11-s − 2·17-s − 4·19-s − 9·25-s + 12·41-s + 26·43-s − 5·49-s + 4·59-s − 4·67-s − 2·73-s + 90·81-s − 8·83-s + 40·89-s + 36·97-s − 72·99-s + 36·107-s + 20·113-s + 7·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 24·153-s + 157-s + ⋯
L(s)  = 1  − 4·9-s + 1.80·11-s − 0.485·17-s − 0.917·19-s − 9/5·25-s + 1.87·41-s + 3.96·43-s − 5/7·49-s + 0.520·59-s − 0.488·67-s − 0.234·73-s + 10·81-s − 0.878·83-s + 4.23·89-s + 3.65·97-s − 7.23·99-s + 3.48·107-s + 1.88·113-s + 7/11·121-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 + 1.94·153-s + 0.0798·157-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{32} \cdot 19^{4}\)
Sign: $1$
Analytic conductor: \(2.27553\times 10^{6}\)
Root analytic conductor: \(6.23211\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{4864} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{32} \cdot 19^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(4.434030955\)
\(L(\frac12)\) \(\approx\) \(4.434030955\)
\(L(\frac{3}{2})\) 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 \( 1 \)
19$C_1$ \( ( 1 + T )^{4} \)
good3$C_2$ \( ( 1 + p T^{2} )^{4} \)
5$C_2^3$ \( 1 + 9 T^{2} + 56 T^{4} + 9 p^{2} T^{6} + p^{4} T^{8} \)
7$C_2^3$ \( 1 + 5 T^{2} - 24 T^{4} + 5 p^{2} T^{6} + p^{4} T^{8} \)
11$D_{4}$ \( ( 1 - 3 T + 10 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \)
13$D_4\times C_2$ \( 1 + 8 T^{2} + 126 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} \)
17$D_{4}$ \( ( 1 + T + 20 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \)
29$D_4\times C_2$ \( 1 + 72 T^{2} + 2750 T^{4} + 72 p^{2} T^{6} + p^{4} T^{8} \)
31$D_4\times C_2$ \( 1 + 80 T^{2} + 3294 T^{4} + 80 p^{2} T^{6} + p^{4} T^{8} \)
37$D_4\times C_2$ \( 1 + 56 T^{2} + 1470 T^{4} + 56 p^{2} T^{6} + p^{4} T^{8} \)
41$D_{4}$ \( ( 1 - 6 T + 34 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
43$D_{4}$ \( ( 1 - 13 T + 114 T^{2} - 13 p T^{3} + p^{2} T^{4} )^{2} \)
47$D_4\times C_2$ \( 1 + 165 T^{2} + 11096 T^{4} + 165 p^{2} T^{6} + p^{4} T^{8} \)
53$C_2^2$ \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \)
59$D_{4}$ \( ( 1 - 2 T + 62 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
61$D_4\times C_2$ \( 1 + 233 T^{2} + 21000 T^{4} + 233 p^{2} T^{6} + p^{4} T^{8} \)
67$D_{4}$ \( ( 1 + 2 T + 78 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
71$D_4\times C_2$ \( 1 - 64 T^{2} + 9054 T^{4} - 64 p^{2} T^{6} + p^{4} T^{8} \)
73$D_{4}$ \( ( 1 + T + 132 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \)
79$D_4\times C_2$ \( 1 + 128 T^{2} + 10878 T^{4} + 128 p^{2} T^{6} + p^{4} T^{8} \)
83$D_{4}$ \( ( 1 + 4 T - 58 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
89$C_2$ \( ( 1 - 10 T + p T^{2} )^{4} \)
97$D_{4}$ \( ( 1 - 18 T + 218 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2} \)
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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

−5.86741181720255623382531897124, −5.82657175494581444939448157838, −5.63181212292909462646689275521, −5.25807034731074097981823525274, −5.24185968637580273279790954416, −4.63719254910866709713489906233, −4.63157610441278403254853045631, −4.47432731601432848201810737571, −4.46377791242291841279008734122, −3.96551048948161181269159641715, −3.64587161388860671380138964950, −3.61799567668676561165421736135, −3.57666372382721906724464424781, −3.20552012658871136346728065822, −2.90465325672932614713043241937, −2.84673044759999219384429092955, −2.47716922343425848702726025959, −2.26537512162627159175577286635, −2.11417243618199858834600937590, −1.88236790733804721724147846657, −1.80592485118276271436380260710, −1.03704251960549542422704917527, −0.61493503151696599195821557926, −0.54598775342365234803748827073, −0.53189352930242743753169850583, 0.53189352930242743753169850583, 0.54598775342365234803748827073, 0.61493503151696599195821557926, 1.03704251960549542422704917527, 1.80592485118276271436380260710, 1.88236790733804721724147846657, 2.11417243618199858834600937590, 2.26537512162627159175577286635, 2.47716922343425848702726025959, 2.84673044759999219384429092955, 2.90465325672932614713043241937, 3.20552012658871136346728065822, 3.57666372382721906724464424781, 3.61799567668676561165421736135, 3.64587161388860671380138964950, 3.96551048948161181269159641715, 4.46377791242291841279008734122, 4.47432731601432848201810737571, 4.63157610441278403254853045631, 4.63719254910866709713489906233, 5.24185968637580273279790954416, 5.25807034731074097981823525274, 5.63181212292909462646689275521, 5.82657175494581444939448157838, 5.86741181720255623382531897124

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