Properties

Label 8-65e8-1.1-c1e4-0-3
Degree $8$
Conductor $3.186\times 10^{14}$
Sign $1$
Analytic cond. $1.29543\times 10^{6}$
Root an. cond. $5.80833$
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  + 4·3-s − 3·4-s − 2·9-s − 12·12-s + 4·16-s − 40·27-s + 6·36-s − 24·43-s + 16·48-s − 22·49-s + 12·53-s − 24·61-s − 9·64-s − 24·79-s − 55·81-s − 36·101-s + 24·103-s − 24·107-s + 120·108-s − 48·113-s − 30·121-s + 127-s − 96·129-s + 131-s + 137-s + 139-s − 8·144-s + ⋯
L(s)  = 1  + 2.30·3-s − 3/2·4-s − 2/3·9-s − 3.46·12-s + 16-s − 7.69·27-s + 36-s − 3.65·43-s + 2.30·48-s − 3.14·49-s + 1.64·53-s − 3.07·61-s − 9/8·64-s − 2.70·79-s − 6.11·81-s − 3.58·101-s + 2.36·103-s − 2.32·107-s + 11.5·108-s − 4.51·113-s − 2.72·121-s + 0.0887·127-s − 8.45·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 2/3·144-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(5^{8} \cdot 13^{8}\)
Sign: $1$
Analytic conductor: \(1.29543\times 10^{6}\)
Root analytic conductor: \(5.80833\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 5^{8} \cdot 13^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

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)$
bad5 \( 1 \)
13 \( 1 \)
good2$C_2^3$ \( 1 + 3 T^{2} + 5 T^{4} + 3 p^{2} T^{6} + p^{4} T^{8} \)
3$C_2$ \( ( 1 - T + p T^{2} )^{4} \)
7$C_2^2$ \( ( 1 + 11 T^{2} + p^{2} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 + 15 T^{2} + p^{2} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 + 13 T^{2} + p^{2} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 + 35 T^{2} + p^{2} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 + 25 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 + 37 T^{2} + p^{2} T^{4} )^{2} \)
31$D_4\times C_2$ \( 1 - 8 T^{2} + 1182 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \)
37$C_2^2$ \( ( 1 + 11 T^{2} + p^{2} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 + 75 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 + 12 T + 101 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
47$D_4\times C_2$ \( 1 + 108 T^{2} + 5990 T^{4} + 108 p^{2} T^{6} + p^{4} T^{8} \)
53$D_{4}$ \( ( 1 - 6 T + 94 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
59$D_4\times C_2$ \( 1 + 30 T^{2} - 1213 T^{4} + 30 p^{2} T^{6} + p^{4} T^{8} \)
61$D_{4}$ \( ( 1 + 12 T + 137 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 + 46 T^{2} - 2589 T^{4} + 46 p^{2} T^{6} + p^{4} T^{8} \)
71$D_4\times C_2$ \( 1 + 222 T^{2} + 22067 T^{4} + 222 p^{2} T^{6} + p^{4} T^{8} \)
73$C_2$ \( ( 1 + p T^{2} )^{4} \)
79$C_2$ \( ( 1 + 6 T + p T^{2} )^{4} \)
83$D_4\times C_2$ \( 1 + 168 T^{2} + 18734 T^{4} + 168 p^{2} T^{6} + p^{4} T^{8} \)
89$D_4\times C_2$ \( 1 + 246 T^{2} + 29627 T^{4} + 246 p^{2} T^{6} + p^{4} T^{8} \)
97$D_4\times C_2$ \( 1 + 238 T^{2} + 29955 T^{4} + 238 p^{2} T^{6} + p^{4} T^{8} \)
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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

−6.40653274144253591804910826330, −6.10790054414509423529655478754, −5.83606618202991594507045657806, −5.70309751905196841787709033739, −5.48367463724769339347063586333, −5.15543189597550309596406538665, −5.07864025059750555498494691380, −5.05998169524061934590686213989, −4.95859399430338701121897177409, −4.25202024765824550805103540326, −4.23733424130308533407960939849, −4.12437277106974833936205662056, −3.94772250365445040374936185751, −3.55314520728420936856347313111, −3.33228885649336475196328454733, −3.29426784704222992747145366046, −3.06949269123369074091378782850, −2.84520654844099799195705295599, −2.74162889031913920510778341787, −2.52431722505783610262443367445, −2.09959358139837587495557312574, −2.06467378330474885516058628763, −1.45166116703540355379810584170, −1.37120413677333864123479405029, −1.27067908942795457637457751669, 0, 0, 0, 0, 1.27067908942795457637457751669, 1.37120413677333864123479405029, 1.45166116703540355379810584170, 2.06467378330474885516058628763, 2.09959358139837587495557312574, 2.52431722505783610262443367445, 2.74162889031913920510778341787, 2.84520654844099799195705295599, 3.06949269123369074091378782850, 3.29426784704222992747145366046, 3.33228885649336475196328454733, 3.55314520728420936856347313111, 3.94772250365445040374936185751, 4.12437277106974833936205662056, 4.23733424130308533407960939849, 4.25202024765824550805103540326, 4.95859399430338701121897177409, 5.05998169524061934590686213989, 5.07864025059750555498494691380, 5.15543189597550309596406538665, 5.48367463724769339347063586333, 5.70309751905196841787709033739, 5.83606618202991594507045657806, 6.10790054414509423529655478754, 6.40653274144253591804910826330

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