Properties

Label 8-3330e4-1.1-c1e4-0-7
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 − 12·11-s + 3·16-s + 20·19-s − 10·25-s + 16·29-s + 8·31-s + 24·44-s + 20·49-s − 8·59-s + 12·61-s − 4·64-s − 8·71-s − 40·76-s − 8·79-s − 48·89-s + 20·100-s + 28·101-s + 12·109-s − 32·116-s + 56·121-s − 16·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + ⋯
L(s)  = 1  − 4-s − 3.61·11-s + 3/4·16-s + 4.58·19-s − 2·25-s + 2.97·29-s + 1.43·31-s + 3.61·44-s + 20/7·49-s − 1.04·59-s + 1.53·61-s − 1/2·64-s − 0.949·71-s − 4.58·76-s − 0.900·79-s − 5.08·89-s + 2·100-s + 2.78·101-s + 1.14·109-s − 2.97·116-s + 5.09·121-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 + ⋯

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

\(L(1)\) \(\approx\) \(2.848017063\)
\(L(\frac12)\) \(\approx\) \(2.848017063\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + T^{2} )^{2} \)
good7$C_2^2$ \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \)
11$C_4$ \( ( 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \)
19$D_{4}$ \( ( 1 - 10 T + 58 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2$ \( ( 1 - 4 T + p T^{2} )^{4} \)
31$C_4$ \( ( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - 60 T^{2} + 1718 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} \)
47$D_4\times C_2$ \( 1 - 80 T^{2} + 4398 T^{4} - 80 p^{2} T^{6} + p^{4} T^{8} \)
53$D_4\times C_2$ \( 1 - 44 T^{2} + 982 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \)
59$D_{4}$ \( ( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
61$D_{4}$ \( ( 1 - 6 T + 126 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 - 156 T^{2} + 12182 T^{4} - 156 p^{2} T^{6} + p^{4} T^{8} \)
71$C_4$ \( ( 1 + 4 T + 126 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 - 180 T^{2} + 15878 T^{4} - 180 p^{2} T^{6} + p^{4} T^{8} \)
79$D_{4}$ \( ( 1 + 4 T + 142 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 + 36 T^{2} + 11222 T^{4} + 36 p^{2} T^{6} + p^{4} T^{8} \)
89$D_{4}$ \( ( 1 + 24 T + 302 T^{2} + 24 p T^{3} + p^{2} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 - 120 T^{2} + 17918 T^{4} - 120 p^{2} T^{6} + p^{4} T^{8} \)
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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

−5.78973490497174609946241737075, −5.78509100599515556184069251776, −5.77052819544889488138933863213, −5.30180452034525172496377265165, −5.26813406530271218581767132734, −5.22357961710516906834780620057, −5.03286306396579850656880247505, −4.75483743473493346713781054592, −4.60321838051821714566620528205, −4.19264849169388590343702379301, −4.16157622452134446446650070368, −3.87423034296310136129149394866, −3.62297148162441168103154543617, −3.24857849580034438004674644354, −3.05209315315871919044350438340, −2.88271444947106895803133639490, −2.65306684101555539982342420554, −2.58306430623257398393348543256, −2.54444247865102832769196945296, −1.86166759101041526326162381819, −1.46539165822552318020526173284, −1.32203384682675710310691758490, −0.837008822747586868315898469466, −0.59691103476414835289244080956, −0.38681883817954171711970255740, 0.38681883817954171711970255740, 0.59691103476414835289244080956, 0.837008822747586868315898469466, 1.32203384682675710310691758490, 1.46539165822552318020526173284, 1.86166759101041526326162381819, 2.54444247865102832769196945296, 2.58306430623257398393348543256, 2.65306684101555539982342420554, 2.88271444947106895803133639490, 3.05209315315871919044350438340, 3.24857849580034438004674644354, 3.62297148162441168103154543617, 3.87423034296310136129149394866, 4.16157622452134446446650070368, 4.19264849169388590343702379301, 4.60321838051821714566620528205, 4.75483743473493346713781054592, 5.03286306396579850656880247505, 5.22357961710516906834780620057, 5.26813406530271218581767132734, 5.30180452034525172496377265165, 5.77052819544889488138933863213, 5.78509100599515556184069251776, 5.78973490497174609946241737075

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