Properties

Label 8-490e4-1.1-c1e4-0-1
Degree $8$
Conductor $57648010000$
Sign $1$
Analytic cond. $234.364$
Root an. cond. $1.97804$
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 − 4·5-s + 3·16-s + 16·19-s + 8·20-s + 8·25-s − 8·29-s − 16·31-s + 24·41-s − 16·59-s − 24·61-s − 4·64-s − 24·71-s − 32·76-s − 8·79-s − 12·80-s − 18·81-s − 40·89-s − 64·95-s − 16·100-s − 24·101-s − 8·109-s + 16·116-s + 4·121-s + 32·124-s − 20·125-s + 127-s + ⋯
L(s)  = 1  − 4-s − 1.78·5-s + 3/4·16-s + 3.67·19-s + 1.78·20-s + 8/5·25-s − 1.48·29-s − 2.87·31-s + 3.74·41-s − 2.08·59-s − 3.07·61-s − 1/2·64-s − 2.84·71-s − 3.67·76-s − 0.900·79-s − 1.34·80-s − 2·81-s − 4.23·89-s − 6.56·95-s − 8/5·100-s − 2.38·101-s − 0.766·109-s + 1.48·116-s + 4/11·121-s + 2.87·124-s − 1.78·125-s + 0.0887·127-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 5^{4} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(234.364\)
Root analytic conductor: \(1.97804\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{490} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 5^{4} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.1397390913\)
\(L(\frac12)\) \(\approx\) \(0.1397390913\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 + T^{2} )^{2} \)
5$C_2^2$ \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
7 \( 1 \)
good3$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \)
13$D_4\times C_2$ \( 1 - 32 T^{2} + 498 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} \)
17$C_2$ \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \)
19$D_{4}$ \( ( 1 - 8 T + 48 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 - 36 T^{2} + 998 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8} \)
29$D_{4}$ \( ( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
31$D_{4}$ \( ( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \)
41$D_{4}$ \( ( 1 - 12 T + 94 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - 92 T^{2} + 4278 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} \)
47$D_4\times C_2$ \( 1 - 108 T^{2} + 5798 T^{4} - 108 p^{2} T^{6} + p^{4} T^{8} \)
53$D_4\times C_2$ \( 1 - 92 T^{2} + 4278 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} \)
59$D_{4}$ \( ( 1 + 8 T + 128 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
61$D_{4}$ \( ( 1 + 12 T + 152 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \)
71$D_{4}$ \( ( 1 + 12 T + 154 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 - 236 T^{2} + 24198 T^{4} - 236 p^{2} T^{6} + p^{4} T^{8} \)
79$D_{4}$ \( ( 1 + 4 T + 138 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 - 160 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2$ \( ( 1 + 10 T + p T^{2} )^{4} \)
97$D_4\times C_2$ \( 1 - 124 T^{2} + 8838 T^{4} - 124 p^{2} T^{6} + p^{4} T^{8} \)
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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

−7.906338874764839013728995370057, −7.53297166757971703566699406231, −7.51815889447922875993865195530, −7.40291975097973064529502854162, −7.19236681637688387315842701221, −6.98123750148877122484900315025, −6.48151124622389409407898717838, −5.95584051141790940636650169868, −5.82524168539910398593475117643, −5.60063092209570976289119478499, −5.55708497476079867512183240825, −5.13906010295543071750474090957, −4.94241521212648658785106116413, −4.32038634891721982666113220059, −4.23042896156416899809120109843, −4.12873622388879084565059835855, −3.98861477662556925968682259245, −3.26684341551782606088953771249, −3.18489074925452591269683327485, −2.96662214029900634387563728569, −2.81336266714014496872473979834, −1.84470633503726856444628060390, −1.28296491791730939813381909729, −1.26772411955896465249025104503, −0.14652777064952203964227770508, 0.14652777064952203964227770508, 1.26772411955896465249025104503, 1.28296491791730939813381909729, 1.84470633503726856444628060390, 2.81336266714014496872473979834, 2.96662214029900634387563728569, 3.18489074925452591269683327485, 3.26684341551782606088953771249, 3.98861477662556925968682259245, 4.12873622388879084565059835855, 4.23042896156416899809120109843, 4.32038634891721982666113220059, 4.94241521212648658785106116413, 5.13906010295543071750474090957, 5.55708497476079867512183240825, 5.60063092209570976289119478499, 5.82524168539910398593475117643, 5.95584051141790940636650169868, 6.48151124622389409407898717838, 6.98123750148877122484900315025, 7.19236681637688387315842701221, 7.40291975097973064529502854162, 7.51815889447922875993865195530, 7.53297166757971703566699406231, 7.906338874764839013728995370057

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