Properties

Label 8-1425e4-1.1-c1e4-0-0
Degree $8$
Conductor $4.123\times 10^{12}$
Sign $1$
Analytic cond. $16763.6$
Root an. cond. $3.37323$
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  + 2·4-s − 2·9-s + 3·16-s − 4·19-s + 8·29-s − 8·31-s − 4·36-s − 24·41-s + 16·49-s + 12·64-s + 16·71-s − 8·76-s + 3·81-s + 8·89-s − 8·101-s − 24·109-s + 16·116-s − 40·121-s − 16·124-s + 127-s + 131-s + 137-s + 139-s − 6·144-s + 149-s + 151-s + 157-s + ⋯
L(s)  = 1  + 4-s − 2/3·9-s + 3/4·16-s − 0.917·19-s + 1.48·29-s − 1.43·31-s − 2/3·36-s − 3.74·41-s + 16/7·49-s + 3/2·64-s + 1.89·71-s − 0.917·76-s + 1/3·81-s + 0.847·89-s − 0.796·101-s − 2.29·109-s + 1.48·116-s − 3.63·121-s − 1.43·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1/2·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{4} \cdot 5^{8} \cdot 19^{4}\)
Sign: $1$
Analytic conductor: \(16763.6\)
Root analytic conductor: \(3.37323\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{4} \cdot 5^{8} \cdot 19^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.2354722561\)
\(L(\frac12)\) \(\approx\) \(0.2354722561\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 + T^{2} )^{2} \)
5 \( 1 \)
19$C_1$ \( ( 1 + T )^{4} \)
good2$D_4\times C_2$ \( 1 - p T^{2} + T^{4} - p^{3} T^{6} + p^{4} T^{8} \)
7$D_4\times C_2$ \( 1 - 16 T^{2} + 130 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \)
11$C_2^2$ \( ( 1 + 20 T^{2} + p^{2} T^{4} )^{2} \)
13$D_4\times C_2$ \( 1 - 16 T^{2} + 274 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \)
17$D_4\times C_2$ \( 1 - 20 T^{2} + 166 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \)
23$D_4\times C_2$ \( 1 - 20 T^{2} + 646 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \)
29$D_{4}$ \( ( 1 - 4 T + 12 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
31$C_4$ \( ( 1 + 4 T - 6 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
37$D_4\times C_2$ \( 1 - 16 T^{2} - 398 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \)
41$D_{4}$ \( ( 1 + 12 T + 116 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - 160 T^{2} + 10066 T^{4} - 160 p^{2} T^{6} + p^{4} T^{8} \)
47$D_4\times C_2$ \( 1 - 52 T^{2} + 486 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \)
53$C_2$ \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2} \)
59$C_2^2$ \( ( 1 + 46 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 + 90 T^{2} + p^{2} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \)
71$D_{4}$ \( ( 1 - 8 T + 86 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 - 142 T^{2} + p^{2} T^{4} )^{2} \)
79$C_2^2$ \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - 180 T^{2} + 20726 T^{4} - 180 p^{2} T^{6} + p^{4} T^{8} \)
89$D_{4}$ \( ( 1 - 4 T + 20 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 - 176 T^{2} + p^{2} T^{4} )^{2} \)
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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

−6.83450472382660994723731136000, −6.51426015492510743831309948680, −6.39455478338696639706121626170, −6.30895510446541431267484849454, −5.99755040317724892408994811045, −5.77783309763764189384947224967, −5.37408684470790528660445431016, −5.28158357826624201873487015586, −5.10304137887260366736252639084, −5.03338798173030763925376351624, −4.57452053714312857723160308504, −4.35048729250825573292645590290, −4.00909195946539138799715523001, −3.68797103428698016705989957426, −3.57875389035724105245981503095, −3.45341836792722375098256444944, −3.10818226997182733522973607819, −2.59785400670633621853230051223, −2.48398024978209773095816678685, −2.24750099196501803270251030260, −2.17208860997407972119271264366, −1.45148502431081731912984660102, −1.36762343570010205407982450013, −0.966363272200579114527837998645, −0.089029927205885183509843580473, 0.089029927205885183509843580473, 0.966363272200579114527837998645, 1.36762343570010205407982450013, 1.45148502431081731912984660102, 2.17208860997407972119271264366, 2.24750099196501803270251030260, 2.48398024978209773095816678685, 2.59785400670633621853230051223, 3.10818226997182733522973607819, 3.45341836792722375098256444944, 3.57875389035724105245981503095, 3.68797103428698016705989957426, 4.00909195946539138799715523001, 4.35048729250825573292645590290, 4.57452053714312857723160308504, 5.03338798173030763925376351624, 5.10304137887260366736252639084, 5.28158357826624201873487015586, 5.37408684470790528660445431016, 5.77783309763764189384947224967, 5.99755040317724892408994811045, 6.30895510446541431267484849454, 6.39455478338696639706121626170, 6.51426015492510743831309948680, 6.83450472382660994723731136000

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