Properties

Label 8-2160e4-1.1-c2e4-0-8
Degree $8$
Conductor $2.177\times 10^{13}$
Sign $1$
Analytic cond. $1.19992\times 10^{7}$
Root an. cond. $7.67174$
Motivic weight $2$
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  + 12·5-s − 12·17-s + 12·19-s − 60·23-s + 66·25-s + 68·31-s + 240·47-s + 8·49-s + 204·53-s − 196·61-s − 180·79-s − 108·83-s − 144·85-s + 144·95-s + 144·107-s − 76·109-s + 48·113-s − 720·115-s + 96·121-s + 156·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 816·155-s + ⋯
L(s)  = 1  + 12/5·5-s − 0.705·17-s + 0.631·19-s − 2.60·23-s + 2.63·25-s + 2.19·31-s + 5.10·47-s + 8/49·49-s + 3.84·53-s − 3.21·61-s − 2.27·79-s − 1.30·83-s − 1.69·85-s + 1.51·95-s + 1.34·107-s − 0.697·109-s + 0.424·113-s − 6.26·115-s + 0.793·121-s + 1.24·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 5.26·155-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 3^{12} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(1.19992\times 10^{7}\)
Root analytic conductor: \(7.67174\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 3^{12} \cdot 5^{4} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(10.87832943\)
\(L(\frac12)\) \(\approx\) \(10.87832943\)
\(L(2)\) 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 \( 1 \)
5$D_{4}$ \( 1 - 12 T + 78 T^{2} - 12 p^{2} T^{3} + p^{4} T^{4} \)
good7$C_2^2 \wr C_2$ \( 1 - 8 T^{2} - 3894 T^{4} - 8 p^{4} T^{6} + p^{8} T^{8} \)
11$C_2^2 \wr C_2$ \( 1 - 96 T^{2} + 29786 T^{4} - 96 p^{4} T^{6} + p^{8} T^{8} \)
13$C_2^2 \wr C_2$ \( 1 - 352 T^{2} + 71898 T^{4} - 352 p^{4} T^{6} + p^{8} T^{8} \)
17$D_{4}$ \( ( 1 + 6 T + 489 T^{2} + 6 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
19$D_{4}$ \( ( 1 - 6 T + 713 T^{2} - 6 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
23$D_{4}$ \( ( 1 + 30 T + 891 T^{2} + 30 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
29$C_2^2 \wr C_2$ \( 1 - 1096 T^{2} + 1242474 T^{4} - 1096 p^{4} T^{6} + p^{8} T^{8} \)
31$D_{4}$ \( ( 1 - 34 T + 1761 T^{2} - 34 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
37$C_2^2 \wr C_2$ \( 1 - 68 T^{2} - 1268634 T^{4} - 68 p^{4} T^{6} + p^{8} T^{8} \)
41$C_2^2 \wr C_2$ \( 1 - 5352 T^{2} + 12407498 T^{4} - 5352 p^{4} T^{6} + p^{8} T^{8} \)
43$C_2^2 \wr C_2$ \( 1 - 5672 T^{2} + 14203050 T^{4} - 5672 p^{4} T^{6} + p^{8} T^{8} \)
47$D_{4}$ \( ( 1 - 120 T + 7626 T^{2} - 120 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
53$D_{4}$ \( ( 1 - 102 T + 8057 T^{2} - 102 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
59$C_2^2 \wr C_2$ \( 1 - 10848 T^{2} + 53616410 T^{4} - 10848 p^{4} T^{6} + p^{8} T^{8} \)
61$D_{4}$ \( ( 1 + 98 T + 8691 T^{2} + 98 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
67$C_2^2 \wr C_2$ \( 1 + 6508 T^{2} + 32310150 T^{4} + 6508 p^{4} T^{6} + p^{8} T^{8} \)
71$C_2^2 \wr C_2$ \( 1 - 4288 T^{2} + 45932730 T^{4} - 4288 p^{4} T^{6} + p^{8} T^{8} \)
73$C_2^2 \wr C_2$ \( 1 - 13280 T^{2} + 84777594 T^{4} - 13280 p^{4} T^{6} + p^{8} T^{8} \)
79$D_{4}$ \( ( 1 + 90 T + 6569 T^{2} + 90 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
83$D_{4}$ \( ( 1 + 54 T + 7779 T^{2} + 54 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
89$C_2^2 \wr C_2$ \( 1 - 8136 T^{2} + 91108874 T^{4} - 8136 p^{4} T^{6} + p^{8} T^{8} \)
97$C_2^2 \wr C_2$ \( 1 - 19172 T^{2} + 267913158 T^{4} - 19172 p^{4} T^{6} + p^{8} T^{8} \)
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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

−6.13774839589161563946079705719, −6.06246853741933668715394170756, −5.92989686215706315473028067388, −5.54322390408332456167204892481, −5.44680284213993208929879238063, −5.41307817540591757693185467611, −5.34042217646076370073087765278, −4.57551437668375885572538932097, −4.43563135722065340416030618874, −4.42531783591077676190979215805, −4.13366944352948936126948494821, −4.12214947735611454171817676491, −3.68859255214530588927471863901, −3.21804391508206378893349653582, −3.16350968335853180262336062290, −2.65947114816755415314769083109, −2.54691348237583543438454584218, −2.43373480886472385655651346187, −2.16910751017547926740051878067, −1.83984433107337903705633117549, −1.73170307781531337687176242200, −1.13757069670227795414694240231, −1.12527015810996137296118657494, −0.60640116098853111664703144823, −0.37044387114129306085824432246, 0.37044387114129306085824432246, 0.60640116098853111664703144823, 1.12527015810996137296118657494, 1.13757069670227795414694240231, 1.73170307781531337687176242200, 1.83984433107337903705633117549, 2.16910751017547926740051878067, 2.43373480886472385655651346187, 2.54691348237583543438454584218, 2.65947114816755415314769083109, 3.16350968335853180262336062290, 3.21804391508206378893349653582, 3.68859255214530588927471863901, 4.12214947735611454171817676491, 4.13366944352948936126948494821, 4.42531783591077676190979215805, 4.43563135722065340416030618874, 4.57551437668375885572538932097, 5.34042217646076370073087765278, 5.41307817540591757693185467611, 5.44680284213993208929879238063, 5.54322390408332456167204892481, 5.92989686215706315473028067388, 6.06246853741933668715394170756, 6.13774839589161563946079705719

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