Properties

Label 8-3330e4-1.1-c1e4-0-0
Degree $8$
Conductor $1.230\times 10^{14}$
Sign $1$
Analytic cond. $499902.$
Root an. cond. $5.15656$
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 + 4·7-s + 3·16-s − 2·25-s − 8·28-s − 20·37-s − 18·49-s − 4·64-s − 8·67-s − 8·73-s + 4·100-s + 12·112-s − 38·121-s + 127-s + 131-s + 137-s + 139-s + 40·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 28·169-s + 173-s − 8·175-s + 179-s + ⋯
L(s)  = 1  − 4-s + 1.51·7-s + 3/4·16-s − 2/5·25-s − 1.51·28-s − 3.28·37-s − 2.57·49-s − 1/2·64-s − 0.977·67-s − 0.936·73-s + 2/5·100-s + 1.13·112-s − 3.45·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 3.28·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2.15·169-s + 0.0760·173-s − 0.604·175-s + 0.0747·179-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 37^{4}\)
Sign: $1$
Analytic conductor: \(499902.\)
Root analytic conductor: \(5.15656\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 37^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.2276003414\)
\(L(\frac12)\) \(\approx\) \(0.2276003414\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 + T^{2} )^{2} \)
3 \( 1 \)
5$C_2$ \( ( 1 + T^{2} )^{2} \)
37$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
good7$C_2$ \( ( 1 - T + p T^{2} )^{4} \)
11$C_2^2$ \( ( 1 + 19 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 - 25 T^{2} + p^{2} T^{4} )^{2} \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \)
23$C_2^2$ \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 49 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 - 35 T^{2} + p^{2} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 + 7 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 - 59 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 + 82 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 + 31 T^{2} + p^{2} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 + 25 T^{2} + p^{2} T^{4} )^{2} \)
67$C_2$ \( ( 1 + 2 T + p T^{2} )^{4} \)
71$C_2^2$ \( ( 1 + 130 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2$ \( ( 1 + 2 T + p T^{2} )^{4} \)
79$C_2^2$ \( ( 1 - 146 T^{2} + p^{2} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 - 191 T^{2} + p^{2} T^{4} )^{2} \)
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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

−5.98089149413964467721915950752, −5.90153129626658521319908454844, −5.80902956566879904455177514603, −5.34190472031009836006322699912, −5.06500581527474188277808596713, −5.03636215328540126409195915737, −4.96956184611027207022036793425, −4.90523915166348422330274639398, −4.46637897319191582248573466564, −4.27015228276851267276479537996, −4.23745890363639284116186280853, −3.71283996181758617006805562221, −3.53285345631304380564586699490, −3.47208465547191561008635861618, −3.46073056943100761626564600361, −2.89569055474120167404705548125, −2.61280996932123977371565166797, −2.42675973114082313757341859551, −2.22705113327365669255729258033, −1.57663535808614299735369875083, −1.56331686439120409303302411622, −1.45389690087854911331172827025, −1.28013417813954279031639402866, −0.51481524909633968431744539134, −0.090901585158609934852298594102, 0.090901585158609934852298594102, 0.51481524909633968431744539134, 1.28013417813954279031639402866, 1.45389690087854911331172827025, 1.56331686439120409303302411622, 1.57663535808614299735369875083, 2.22705113327365669255729258033, 2.42675973114082313757341859551, 2.61280996932123977371565166797, 2.89569055474120167404705548125, 3.46073056943100761626564600361, 3.47208465547191561008635861618, 3.53285345631304380564586699490, 3.71283996181758617006805562221, 4.23745890363639284116186280853, 4.27015228276851267276479537996, 4.46637897319191582248573466564, 4.90523915166348422330274639398, 4.96956184611027207022036793425, 5.03636215328540126409195915737, 5.06500581527474188277808596713, 5.34190472031009836006322699912, 5.80902956566879904455177514603, 5.90153129626658521319908454844, 5.98089149413964467721915950752

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