Properties

Label 8-1560e4-1.1-c1e4-0-5
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·3-s + 10·9-s + 8·13-s + 14·17-s − 2·23-s − 2·25-s − 20·27-s + 8·29-s − 32·39-s + 20·43-s + 3·49-s − 56·51-s + 22·53-s + 14·61-s + 8·69-s + 8·75-s − 26·79-s + 35·81-s − 32·87-s + 16·101-s + 12·103-s + 14·107-s + 56·113-s + 80·117-s + 23·121-s + 127-s − 80·129-s + ⋯
L(s)  = 1  − 2.30·3-s + 10/3·9-s + 2.21·13-s + 3.39·17-s − 0.417·23-s − 2/5·25-s − 3.84·27-s + 1.48·29-s − 5.12·39-s + 3.04·43-s + 3/7·49-s − 7.84·51-s + 3.02·53-s + 1.79·61-s + 0.963·69-s + 0.923·75-s − 2.92·79-s + 35/9·81-s − 3.43·87-s + 1.59·101-s + 1.18·103-s + 1.35·107-s + 5.26·113-s + 7.39·117-s + 2.09·121-s + 0.0887·127-s − 7.04·129-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

\(L(1)\) \(\approx\) \(3.425139415\)
\(L(\frac12)\) \(\approx\) \(3.425139415\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_1$ \( ( 1 + T )^{4} \)
5$C_2$ \( ( 1 + T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
good7$D_4\times C_2$ \( 1 - 3 T^{2} + 8 T^{4} - 3 p^{2} T^{6} + p^{4} T^{8} \)
11$D_4\times C_2$ \( 1 - 23 T^{2} + 364 T^{4} - 23 p^{2} T^{6} + p^{4} T^{8} \)
17$D_{4}$ \( ( 1 - 7 T + 36 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \)
19$D_4\times C_2$ \( 1 + 8 T^{2} + 574 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} \)
23$D_{4}$ \( ( 1 + T + 36 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{4} \)
31$D_4\times C_2$ \( 1 - 24 T^{2} + 590 T^{4} - 24 p^{2} T^{6} + p^{4} T^{8} \)
37$D_4\times C_2$ \( 1 - 127 T^{2} + 6760 T^{4} - 127 p^{2} T^{6} + p^{4} T^{8} \)
41$D_4\times C_2$ \( 1 - 83 T^{2} + 3844 T^{4} - 83 p^{2} T^{6} + p^{4} T^{8} \)
43$D_{4}$ \( ( 1 - 10 T + 70 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{2} \)
53$D_{4}$ \( ( 1 - 11 T + 126 T^{2} - 11 p T^{3} + p^{2} T^{4} )^{2} \)
59$D_4\times C_2$ \( 1 - 56 T^{2} - 290 T^{4} - 56 p^{2} T^{6} + p^{4} T^{8} \)
61$D_{4}$ \( ( 1 - 7 T + 124 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \)
71$D_4\times C_2$ \( 1 - 251 T^{2} + 25576 T^{4} - 251 p^{2} T^{6} + p^{4} T^{8} \)
73$C_2^2$ \( ( 1 + 18 T^{2} + p^{2} T^{4} )^{2} \)
79$D_{4}$ \( ( 1 + 13 T + 190 T^{2} + 13 p T^{3} + p^{2} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - 232 T^{2} + 25758 T^{4} - 232 p^{2} T^{6} + p^{4} T^{8} \)
89$D_4\times C_2$ \( 1 - 331 T^{2} + 43140 T^{4} - 331 p^{2} T^{6} + p^{4} T^{8} \)
97$D_4\times C_2$ \( 1 - 119 T^{2} + 6768 T^{4} - 119 p^{2} T^{6} + p^{4} T^{8} \)
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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

−6.67136046167502049795790585488, −6.16218576945072095423819909918, −6.14409439771898204304702971522, −6.04263918642760352340893823644, −6.02346402499296233392244566944, −5.68289125802599874581345888962, −5.52434472409603220782377137220, −5.25345743083173724768907328748, −5.17602536360874176620265530429, −4.91388579475347018628894173506, −4.40495835501734995547379675448, −4.39246843071989839539984933674, −4.04073763065268806331557545790, −3.80677124860558572738620801790, −3.79222527395488031424655847036, −3.31578162928070297105122985696, −3.20602186740171614901742670909, −2.79417707023765875861947110796, −2.45333298729907874267942337771, −2.02746551029165210818526086140, −1.74885985166863053704684536533, −1.11805652041940596318669843942, −1.10347828958971854439934056747, −0.795757558177310728955073776668, −0.63303209551320106903255308201, 0.63303209551320106903255308201, 0.795757558177310728955073776668, 1.10347828958971854439934056747, 1.11805652041940596318669843942, 1.74885985166863053704684536533, 2.02746551029165210818526086140, 2.45333298729907874267942337771, 2.79417707023765875861947110796, 3.20602186740171614901742670909, 3.31578162928070297105122985696, 3.79222527395488031424655847036, 3.80677124860558572738620801790, 4.04073763065268806331557545790, 4.39246843071989839539984933674, 4.40495835501734995547379675448, 4.91388579475347018628894173506, 5.17602536360874176620265530429, 5.25345743083173724768907328748, 5.52434472409603220782377137220, 5.68289125802599874581345888962, 6.02346402499296233392244566944, 6.04263918642760352340893823644, 6.14409439771898204304702971522, 6.16218576945072095423819909918, 6.67136046167502049795790585488

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