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Properties

Label 8-39e8-1.1-c1e4-0-5
Degree $8$
Conductor $5.352\times 10^{12}$
Sign $1$
Analytic cond. $21758.3$
Root an. cond. $3.48500$
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  + 5·16-s + 16·43-s − 28·49-s + 64·103-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + ⋯
L(s)  = 1  + 5/4·16-s + 2.43·43-s − 4·49-s + 6.30·103-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 + 0.0773·167-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + 0.0655·233-s + 0.0646·239-s + 0.0644·241-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{8} \cdot 13^{8}\)
Sign: $1$
Analytic conductor: \(21758.3\)
Root analytic conductor: \(3.48500\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{8} \cdot 13^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(4.080066574\)
\(L(\frac12)\) \(\approx\) \(4.080066574\)
\(L(\frac{3}{2})\) 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 \( 1 \)
13 \( 1 \)
good2$D_4\times C_2$ \( 1 - 5 T^{4} + p^{4} T^{8} \)
5$D_4\times C_2$ \( 1 - 2 T^{4} + p^{4} T^{8} \)
7$C_2$ \( ( 1 + p T^{2} )^{4} \)
11$D_4\times C_2$ \( 1 + 190 T^{4} + p^{4} T^{8} \)
17$C_2$ \( ( 1 + p T^{2} )^{4} \)
19$C_2$ \( ( 1 + p T^{2} )^{4} \)
23$C_2$ \( ( 1 + p T^{2} )^{4} \)
29$C_2$ \( ( 1 + p T^{2} )^{4} \)
31$C_2$ \( ( 1 + p T^{2} )^{4} \)
37$C_2$ \( ( 1 + p T^{2} )^{4} \)
41$D_4\times C_2$ \( 1 - 2930 T^{4} + p^{4} T^{8} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{4} \)
47$D_4\times C_2$ \( 1 - 4370 T^{4} + p^{4} T^{8} \)
53$C_2$ \( ( 1 + p T^{2} )^{4} \)
59$D_4\times C_2$ \( 1 + 6910 T^{4} + p^{4} T^{8} \)
61$C_2^2$ \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \)
67$C_2$ \( ( 1 + p T^{2} )^{4} \)
71$D_4\times C_2$ \( 1 + 3790 T^{4} + p^{4} T^{8} \)
73$C_2$ \( ( 1 + p T^{2} )^{4} \)
79$C_2^2$ \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - 13730 T^{4} + p^{4} T^{8} \)
89$D_4\times C_2$ \( 1 + 9550 T^{4} + p^{4} T^{8} \)
97$C_2$ \( ( 1 + p T^{2} )^{4} \)
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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

−6.72759103535354790310584575355, −6.59281493847754866318962329691, −6.26407300078948751418197268499, −6.03178352607861490453876258267, −5.86097616307762742823496354415, −5.84195345003448296942681293379, −5.37516921414065477244529195254, −5.33009218887665442804051082088, −5.01122673888715230047059768815, −4.57330097361008825381055794427, −4.56456954118016983866394286818, −4.45908192042152096884951322521, −4.16249533876605890771860230802, −3.57862380011819886478115676700, −3.50525278096381922966679599021, −3.42785021766152012935178947781, −3.09332773808433018621089286874, −2.79241847977280294212214510126, −2.62308738013551878535473690468, −1.96680969462411954523886484275, −1.86984365136676049867959801908, −1.77963582417322859946651968903, −1.14339478440729767047013499426, −0.72133101212585535387029658531, −0.50365589663493428295635763476, 0.50365589663493428295635763476, 0.72133101212585535387029658531, 1.14339478440729767047013499426, 1.77963582417322859946651968903, 1.86984365136676049867959801908, 1.96680969462411954523886484275, 2.62308738013551878535473690468, 2.79241847977280294212214510126, 3.09332773808433018621089286874, 3.42785021766152012935178947781, 3.50525278096381922966679599021, 3.57862380011819886478115676700, 4.16249533876605890771860230802, 4.45908192042152096884951322521, 4.56456954118016983866394286818, 4.57330097361008825381055794427, 5.01122673888715230047059768815, 5.33009218887665442804051082088, 5.37516921414065477244529195254, 5.84195345003448296942681293379, 5.86097616307762742823496354415, 6.03178352607861490453876258267, 6.26407300078948751418197268499, 6.59281493847754866318962329691, 6.72759103535354790310584575355

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