Properties

Label 8-504e4-1.1-c1e4-0-9
Degree $8$
Conductor $64524128256$
Sign $1$
Analytic cond. $262.319$
Root an. cond. $2.00610$
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 + 16·25-s + 8·28-s + 8·31-s + 10·49-s − 8·64-s + 8·73-s + 32·79-s − 40·97-s + 32·100-s + 8·103-s + 40·121-s + 16·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 52·169-s + 173-s + 64·175-s + 179-s + ⋯
L(s)  = 1  + 4-s + 1.51·7-s + 16/5·25-s + 1.51·28-s + 1.43·31-s + 10/7·49-s − 64-s + 0.936·73-s + 3.60·79-s − 4.06·97-s + 16/5·100-s + 0.788·103-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 + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 4·169-s + 0.0760·173-s + 4.83·175-s + 0.0747·179-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{12} \cdot 3^{8} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(262.319\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{12} \cdot 3^{8} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(4.909398369\)
\(L(\frac12)\) \(\approx\) \(4.909398369\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
3 \( 1 \)
7$C_1$ \( ( 1 - T )^{4} \)
good5$C_2^2$ \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2$ \( ( 1 - p T^{2} )^{4} \)
17$C_2^2$ \( ( 1 + 28 T^{2} + p^{2} T^{4} )^{2} \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \)
23$C_2^2$ \( ( 1 + 40 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2$ \( ( 1 - 2 T + p T^{2} )^{4} \)
37$C_2^2$ \( ( 1 - 62 T^{2} + p^{2} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 + 28 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \)
47$C_2^2$ \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \)
67$C_2$ \( ( 1 - 16 T + p T^{2} )^{2}( 1 + 16 T + p T^{2} )^{2} \)
71$C_2^2$ \( ( 1 + 88 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{4} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{4} \)
83$C_2$ \( ( 1 - 18 T + p T^{2} )^{2}( 1 + 18 T + p T^{2} )^{2} \)
89$C_2^2$ \( ( 1 + 28 T^{2} + p^{2} T^{4} )^{2} \)
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{4} \)
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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

−8.050535525008617212301035740878, −7.62395382306754769871625524029, −7.37900434091854391118146853593, −7.09702954877581353712381914170, −6.78347985266239099448637801591, −6.75669555127918664552439288560, −6.71650888863171536925222881926, −6.18657952027317844392775494797, −5.83002641205238675019185268288, −5.79496995979326825796546263568, −5.41732281679631057215162647542, −4.88488549499706473560029556004, −4.87223307419506817932320292005, −4.67390593224415943310808325140, −4.59216245939074463375673807545, −3.99966843890866509262109421250, −3.66858830125814669163406463903, −3.38340487141076566696957292529, −2.98741242597082494602617776280, −2.66300379041176097578364095511, −2.29771094923668618910981038416, −2.23893208315553379262902045330, −1.52482990367394484839681975845, −1.20982805244815938810505320924, −0.806470209863134182034020250678, 0.806470209863134182034020250678, 1.20982805244815938810505320924, 1.52482990367394484839681975845, 2.23893208315553379262902045330, 2.29771094923668618910981038416, 2.66300379041176097578364095511, 2.98741242597082494602617776280, 3.38340487141076566696957292529, 3.66858830125814669163406463903, 3.99966843890866509262109421250, 4.59216245939074463375673807545, 4.67390593224415943310808325140, 4.87223307419506817932320292005, 4.88488549499706473560029556004, 5.41732281679631057215162647542, 5.79496995979326825796546263568, 5.83002641205238675019185268288, 6.18657952027317844392775494797, 6.71650888863171536925222881926, 6.75669555127918664552439288560, 6.78347985266239099448637801591, 7.09702954877581353712381914170, 7.37900434091854391118146853593, 7.62395382306754769871625524029, 8.050535525008617212301035740878

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