Properties

Label 8-1560e4-1.1-c1e4-0-1
Degree $8$
Conductor $5.922\times 10^{12}$
Sign $1$
Analytic cond. $24077.2$
Root an. cond. $3.52939$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 2·4-s + 12·7-s + 4·8-s − 2·9-s + 24·14-s + 8·16-s − 16·17-s − 4·18-s − 8·23-s − 2·25-s + 24·28-s − 20·31-s + 8·32-s − 32·34-s − 4·36-s + 8·41-s − 16·46-s − 28·47-s + 68·49-s − 4·50-s + 48·56-s − 40·62-s − 24·63-s + 8·64-s − 32·68-s + 60·71-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s + 4.53·7-s + 1.41·8-s − 2/3·9-s + 6.41·14-s + 2·16-s − 3.88·17-s − 0.942·18-s − 1.66·23-s − 2/5·25-s + 4.53·28-s − 3.59·31-s + 1.41·32-s − 5.48·34-s − 2/3·36-s + 1.24·41-s − 2.35·46-s − 4.08·47-s + 68/7·49-s − 0.565·50-s + 6.41·56-s − 5.08·62-s − 3.02·63-s + 64-s − 3.88·68-s + 7.12·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{12} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(24077.2\)
Root analytic conductor: \(3.52939\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{12} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(6.978597855\)
\(L(\frac12)\) \(\approx\) \(6.978597855\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} \)
3$C_2$ \( ( 1 + T^{2} )^{2} \)
5$C_2$ \( ( 1 + T^{2} )^{2} \)
13$C_2$ \( ( 1 + T^{2} )^{2} \)
good7$D_{4}$ \( ( 1 - 6 T + 20 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
11$D_4\times C_2$ \( 1 - 20 T^{2} + 234 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \)
17$C_2$ \( ( 1 + 4 T + p T^{2} )^{4} \)
19$D_4\times C_2$ \( 1 - 68 T^{2} + 1866 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} \)
23$C_2$ \( ( 1 + 2 T + p T^{2} )^{4} \)
29$D_4\times C_2$ \( 1 - 12 T^{2} + 950 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \)
31$D_{4}$ \( ( 1 + 10 T + 60 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \)
37$D_4\times C_2$ \( 1 - 92 T^{2} + 4086 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} \)
41$D_{4}$ \( ( 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - 140 T^{2} + 8406 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \)
47$D_{4}$ \( ( 1 + 14 T + 140 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \)
53$D_4\times C_2$ \( 1 + 12 T^{2} + 3926 T^{4} + 12 p^{2} T^{6} + p^{4} T^{8} \)
59$D_4\times C_2$ \( 1 - 84 T^{2} + 8426 T^{4} - 84 p^{2} T^{6} + p^{4} T^{8} \)
61$D_4\times C_2$ \( 1 - 116 T^{2} + 7734 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} \)
67$D_4\times C_2$ \( 1 - 116 T^{2} + 7050 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} \)
71$D_{4}$ \( ( 1 - 30 T + 364 T^{2} - 30 p T^{3} + p^{2} T^{4} )^{2} \)
73$D_{4}$ \( ( 1 - 4 T - 42 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
79$D_{4}$ \( ( 1 - 28 T + 342 T^{2} - 28 p T^{3} + p^{2} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - 228 T^{2} + 26186 T^{4} - 228 p^{2} T^{6} + p^{4} T^{8} \)
89$D_{4}$ \( ( 1 + 12 T + 166 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
97$D_{4}$ \( ( 1 + 8 T + 198 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−6.69135419388266077654324059130, −6.47330423905953804782717938067, −6.46385023775830908962635605748, −5.88629558276592332916677587575, −5.65750417025560596646319653681, −5.41391005453253864284616807636, −5.19790737021248705523940474248, −5.18015017429895660334714508170, −5.08654735174228715226036359524, −4.71198011767661213334661633230, −4.41051578980425848046937070899, −4.39960246930139254560480992651, −4.38831207021029009336823300371, −3.77060443452756139393131604342, −3.76957795076246839035356386999, −3.51618001337910201092167483964, −3.28479381709122669931245149878, −2.32709730182347794497301344342, −2.29861345109544461781731957654, −2.13956667825070392119853210297, −1.98705410388537603673679862340, −1.91267719917963710996793127574, −1.42999794106615314411448265830, −1.14311860999950698266738182552, −0.30578429124910456884843301090, 0.30578429124910456884843301090, 1.14311860999950698266738182552, 1.42999794106615314411448265830, 1.91267719917963710996793127574, 1.98705410388537603673679862340, 2.13956667825070392119853210297, 2.29861345109544461781731957654, 2.32709730182347794497301344342, 3.28479381709122669931245149878, 3.51618001337910201092167483964, 3.76957795076246839035356386999, 3.77060443452756139393131604342, 4.38831207021029009336823300371, 4.39960246930139254560480992651, 4.41051578980425848046937070899, 4.71198011767661213334661633230, 5.08654735174228715226036359524, 5.18015017429895660334714508170, 5.19790737021248705523940474248, 5.41391005453253864284616807636, 5.65750417025560596646319653681, 5.88629558276592332916677587575, 6.46385023775830908962635605748, 6.47330423905953804782717938067, 6.69135419388266077654324059130

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