Properties

Label 8-2475e4-1.1-c1e4-0-9
Degree $8$
Conductor $3.752\times 10^{13}$
Sign $1$
Analytic cond. $152548.$
Root an. cond. $4.44555$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 5·4-s − 4·11-s + 12·16-s − 10·29-s − 12·31-s + 12·41-s − 20·44-s + 5·49-s + 20·59-s − 22·61-s + 15·64-s + 12·71-s − 10·79-s − 50·89-s + 2·101-s − 10·109-s − 50·116-s + 10·121-s − 60·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯
L(s)  = 1  + 5/2·4-s − 1.20·11-s + 3·16-s − 1.85·29-s − 2.15·31-s + 1.87·41-s − 3.01·44-s + 5/7·49-s + 2.60·59-s − 2.81·61-s + 15/8·64-s + 1.42·71-s − 1.12·79-s − 5.29·89-s + 0.199·101-s − 0.957·109-s − 4.64·116-s + 0.909·121-s − 5.38·124-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{8} \cdot 5^{8} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(152548.\)
Root analytic conductor: \(4.44555\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{8} \cdot 5^{8} \cdot 11^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(6.382800595\)
\(L(\frac12)\) \(\approx\) \(6.382800595\)
\(L(\frac{3}{2})\) 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)$
bad3 \( 1 \)
5 \( 1 \)
11$C_1$ \( ( 1 + T )^{4} \)
good2$D_4\times C_2$ \( 1 - 5 T^{2} + 13 T^{4} - 5 p^{2} T^{6} + p^{4} T^{8} \)
7$D_4\times C_2$ \( 1 - 5 T^{2} + 93 T^{4} - 5 p^{2} T^{6} + p^{4} T^{8} \)
13$D_4\times C_2$ \( 1 - 10 T^{2} + 43 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} \)
17$D_4\times C_2$ \( 1 - 65 T^{2} + 1633 T^{4} - 65 p^{2} T^{6} + p^{4} T^{8} \)
19$C_2^2$ \( ( 1 - 7 T^{2} + p^{2} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 - 25 T^{2} + 933 T^{4} - 25 p^{2} T^{6} + p^{4} T^{8} \)
29$D_{4}$ \( ( 1 + 5 T + 63 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \)
31$C_2$ \( ( 1 + 3 T + p T^{2} )^{4} \)
37$D_4\times C_2$ \( 1 - 10 T^{2} + 1483 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} \)
41$C_2$ \( ( 1 - 3 T + p T^{2} )^{4} \)
43$C_2^2$ \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \)
47$D_4\times C_2$ \( 1 - 10 T^{2} + 1563 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} \)
53$D_4\times C_2$ \( 1 - 85 T^{2} + 6873 T^{4} - 85 p^{2} T^{6} + p^{4} T^{8} \)
59$D_{4}$ \( ( 1 - 10 T + 123 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \)
61$D_{4}$ \( ( 1 + 11 T + 121 T^{2} + 11 p T^{3} + p^{2} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \)
71$C_4$ \( ( 1 - 6 T + 26 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 - 25 T^{2} + 10153 T^{4} - 25 p^{2} T^{6} + p^{4} T^{8} \)
79$D_{4}$ \( ( 1 + 5 T + 153 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 + 55 T^{2} + 6333 T^{4} + 55 p^{2} T^{6} + p^{4} T^{8} \)
89$D_{4}$ \( ( 1 + 25 T + 303 T^{2} + 25 p T^{3} + p^{2} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 - 385 T^{2} + 55873 T^{4} - 385 p^{2} T^{6} + p^{4} T^{8} \)
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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

−6.25140904766786369078729666871, −6.11420726679279898764888689439, −6.02451457361668432536057251976, −5.63580742160780555687085883782, −5.57809095556498022714021825097, −5.52594788836928962383101823468, −5.16364347829369655213113917529, −5.04555203664522979998740174837, −4.77777673164639708579564786536, −4.22457129521533049983783528263, −4.13968070942314629719524474459, −3.94398583472337731991702966750, −3.90379843168599400456502953223, −3.35639636267667604518408838064, −3.28379230514534594698192230935, −2.80813566977082935146137594383, −2.64305368563454441458485811318, −2.59408464708222606013377943235, −2.53245702393390904258852037015, −1.76889099742436147182389627533, −1.76240417828612586298479929448, −1.74494769094619936449408969549, −1.38568465391928222869429663086, −0.54091982965008847509965374485, −0.46438049348119402901010710630, 0.46438049348119402901010710630, 0.54091982965008847509965374485, 1.38568465391928222869429663086, 1.74494769094619936449408969549, 1.76240417828612586298479929448, 1.76889099742436147182389627533, 2.53245702393390904258852037015, 2.59408464708222606013377943235, 2.64305368563454441458485811318, 2.80813566977082935146137594383, 3.28379230514534594698192230935, 3.35639636267667604518408838064, 3.90379843168599400456502953223, 3.94398583472337731991702966750, 4.13968070942314629719524474459, 4.22457129521533049983783528263, 4.77777673164639708579564786536, 5.04555203664522979998740174837, 5.16364347829369655213113917529, 5.52594788836928962383101823468, 5.57809095556498022714021825097, 5.63580742160780555687085883782, 6.02451457361668432536057251976, 6.11420726679279898764888689439, 6.25140904766786369078729666871

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