Properties

Label 8-1400e4-1.1-c1e4-0-3
Degree $8$
Conductor $3.842\times 10^{12}$
Sign $1$
Analytic cond. $15617.8$
Root an. cond. $3.34350$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 5·9-s + 14·11-s − 4·19-s + 6·29-s − 32·31-s + 4·41-s − 2·49-s − 32·59-s + 12·61-s + 32·71-s − 26·79-s + 9·81-s − 36·89-s − 70·99-s − 24·101-s + 10·109-s + 95·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 31·169-s + ⋯
L(s)  = 1  − 5/3·9-s + 4.22·11-s − 0.917·19-s + 1.11·29-s − 5.74·31-s + 0.624·41-s − 2/7·49-s − 4.16·59-s + 1.53·61-s + 3.79·71-s − 2.92·79-s + 81-s − 3.81·89-s − 7.03·99-s − 2.38·101-s + 0.957·109-s + 8.63·121-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 + 2.38·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{12} \cdot 5^{8} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(15617.8\)
Root analytic conductor: \(3.34350\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1400} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{12} \cdot 5^{8} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.6784598576\)
\(L(\frac12)\) \(\approx\) \(0.6784598576\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
7$C_2$ \( ( 1 + T^{2} )^{2} \)
good3$C_2^3$ \( 1 + 5 T^{2} + 16 T^{4} + 5 p^{2} T^{6} + p^{4} T^{8} \)
11$D_{4}$ \( ( 1 - 7 T + 26 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \)
13$D_4\times C_2$ \( 1 - 31 T^{2} + 504 T^{4} - 31 p^{2} T^{6} + p^{4} T^{8} \)
17$D_4\times C_2$ \( 1 - 39 T^{2} + 752 T^{4} - 39 p^{2} T^{6} + p^{4} T^{8} \)
19$D_{4}$ \( ( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 - 24 T^{2} + 1070 T^{4} - 24 p^{2} T^{6} + p^{4} T^{8} \)
29$D_{4}$ \( ( 1 - 3 T + 52 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \)
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{4} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \)
41$D_{4}$ \( ( 1 - 2 T + 50 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - 88 T^{2} + 4446 T^{4} - 88 p^{2} T^{6} + p^{4} T^{8} \)
47$D_4\times C_2$ \( 1 - 35 T^{2} + 4056 T^{4} - 35 p^{2} T^{6} + p^{4} T^{8} \)
53$D_4\times C_2$ \( 1 - 96 T^{2} + 4622 T^{4} - 96 p^{2} T^{6} + p^{4} T^{8} \)
59$C_2$ \( ( 1 + 8 T + p T^{2} )^{4} \)
61$D_{4}$ \( ( 1 - 6 T + 98 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 - 118 T^{2} + p^{2} T^{4} )^{2} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{4} \)
73$C_2$ \( ( 1 - 16 T + p T^{2} )^{2}( 1 + 16 T + p T^{2} )^{2} \)
79$D_{4}$ \( ( 1 + 13 T + 126 T^{2} + 13 p T^{3} + p^{2} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - 60 T^{2} + 12566 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} \)
89$D_{4}$ \( ( 1 + 18 T + 226 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 + 65 T^{2} + 3168 T^{4} + 65 p^{2} T^{6} + p^{4} T^{8} \)
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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

−6.77517282403064241343994874454, −6.58114040421040890928974585549, −6.47065943547911520903316357830, −6.19361746890550661782612452121, −5.86976570282921137780113610587, −5.84558205267378469929778189905, −5.59314833508952772011714162803, −5.36300106248411565105561122905, −5.18028005374838866352458891943, −4.81562663298553157029220312574, −4.44244714552740709788916182633, −4.13403506249881223823261885110, −4.09104890284744200222368086986, −3.98835724350761741555531399266, −3.57897200567973441117360518701, −3.47034624048576921377016098199, −3.07609754494843079586119052947, −3.04084918377768325476006952003, −2.42422320143556882085923600910, −2.24573832105176853854466750475, −1.63614467732179163470468212709, −1.58908731437478668418211975694, −1.51379402837320561826136960514, −0.878826684662716333999914239061, −0.16691496737015332320128859463, 0.16691496737015332320128859463, 0.878826684662716333999914239061, 1.51379402837320561826136960514, 1.58908731437478668418211975694, 1.63614467732179163470468212709, 2.24573832105176853854466750475, 2.42422320143556882085923600910, 3.04084918377768325476006952003, 3.07609754494843079586119052947, 3.47034624048576921377016098199, 3.57897200567973441117360518701, 3.98835724350761741555531399266, 4.09104890284744200222368086986, 4.13403506249881223823261885110, 4.44244714552740709788916182633, 4.81562663298553157029220312574, 5.18028005374838866352458891943, 5.36300106248411565105561122905, 5.59314833508952772011714162803, 5.84558205267378469929778189905, 5.86976570282921137780113610587, 6.19361746890550661782612452121, 6.47065943547911520903316357830, 6.58114040421040890928974585549, 6.77517282403064241343994874454

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