Properties

Label 8-1400e4-1.1-c1e4-0-5
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  + 3·9-s − 2·11-s + 12·19-s − 10·29-s − 8·31-s + 12·41-s − 2·49-s + 16·59-s − 44·61-s + 22·79-s − 7·81-s − 4·89-s − 6·99-s + 24·101-s + 26·109-s − 33·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 25·169-s + 36·171-s + ⋯
L(s)  = 1  + 9-s − 0.603·11-s + 2.75·19-s − 1.85·29-s − 1.43·31-s + 1.87·41-s − 2/7·49-s + 2.08·59-s − 5.63·61-s + 2.47·79-s − 7/9·81-s − 0.423·89-s − 0.603·99-s + 2.38·101-s + 2.49·109-s − 3·121-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 − 1.92·169-s + 2.75·171-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

\(L(1)\) \(\approx\) \(2.231791648\)
\(L(\frac12)\) \(\approx\) \(2.231791648\)
\(L(\frac{3}{2})\) 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$D_4\times C_2$ \( 1 - p T^{2} + 16 T^{4} - p^{3} T^{6} + p^{4} T^{8} \)
11$D_{4}$ \( ( 1 + T + 18 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \)
13$C_2^3$ \( 1 + 25 T^{2} + 456 T^{4} + 25 p^{2} T^{6} + p^{4} T^{8} \)
17$D_4\times C_2$ \( 1 + T^{2} + 64 T^{4} + p^{2} T^{6} + p^{4} T^{8} \)
19$D_{4}$ \( ( 1 - 6 T + 30 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 - 56 T^{2} + 1774 T^{4} - 56 p^{2} T^{6} + p^{4} T^{8} \)
29$D_{4}$ \( ( 1 + 5 T + 60 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \)
31$D_{4}$ \( ( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \)
41$D_{4}$ \( ( 1 - 6 T + 74 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - 120 T^{2} + 6686 T^{4} - 120 p^{2} T^{6} + p^{4} T^{8} \)
47$D_4\times C_2$ \( 1 - 139 T^{2} + 8904 T^{4} - 139 p^{2} T^{6} + p^{4} T^{8} \)
53$D_4\times C_2$ \( 1 - 16 T^{2} + 174 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{4} \)
61$D_{4}$ \( ( 1 + 22 T + 226 T^{2} + 22 p T^{3} + p^{2} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 - 60 T^{2} + 86 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} \)
71$C_2$ \( ( 1 + p T^{2} )^{4} \)
73$D_4\times C_2$ \( 1 - 124 T^{2} + 10150 T^{4} - 124 p^{2} T^{6} + p^{4} T^{8} \)
79$D_{4}$ \( ( 1 - 11 T + 150 T^{2} - 11 p T^{3} + p^{2} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \)
89$D_{4}$ \( ( 1 + 2 T + 162 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 - 199 T^{2} + 20112 T^{4} - 199 p^{2} T^{6} + p^{4} T^{8} \)
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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

−7.01129556534427408274517385793, −6.51148482547309782437921226605, −6.40493314811282025128482289080, −6.16134918238568241350375035505, −5.91100423073670789324109130592, −5.64975068822626243874603517801, −5.55387426926461983209516831355, −5.27824651757590745895446837867, −5.18986133889842048722558690371, −4.75128985922686082330200039107, −4.59427708723495744443373766246, −4.36550296875690537420777411144, −4.26622302712820524244539116873, −3.68352557202805543211790362009, −3.49295853126026088006959778651, −3.46915940855910231562766083746, −3.21512375716778850391782945953, −2.85968825103036031012714762032, −2.51305106345929472698487532338, −2.16665484042412804600039542478, −1.93643804931873142741763856806, −1.57004024807407151026381495170, −1.21085809393547864215875895238, −0.951819091804065375769124304012, −0.30958258985644181039084765392, 0.30958258985644181039084765392, 0.951819091804065375769124304012, 1.21085809393547864215875895238, 1.57004024807407151026381495170, 1.93643804931873142741763856806, 2.16665484042412804600039542478, 2.51305106345929472698487532338, 2.85968825103036031012714762032, 3.21512375716778850391782945953, 3.46915940855910231562766083746, 3.49295853126026088006959778651, 3.68352557202805543211790362009, 4.26622302712820524244539116873, 4.36550296875690537420777411144, 4.59427708723495744443373766246, 4.75128985922686082330200039107, 5.18986133889842048722558690371, 5.27824651757590745895446837867, 5.55387426926461983209516831355, 5.64975068822626243874603517801, 5.91100423073670789324109130592, 6.16134918238568241350375035505, 6.40493314811282025128482289080, 6.51148482547309782437921226605, 7.01129556534427408274517385793

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