Properties

Label 8-2646e4-1.1-c1e4-0-3
Degree $8$
Conductor $4.902\times 10^{13}$
Sign $1$
Analytic cond. $199281.$
Root an. cond. $4.59656$
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  − 2·4-s + 3·16-s − 20·25-s − 32·37-s + 16·43-s − 4·64-s + 8·67-s − 52·79-s + 40·100-s − 32·109-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 64·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 28·169-s − 32·172-s + 173-s + 179-s + 181-s + 191-s + ⋯
L(s)  = 1  − 4-s + 3/4·16-s − 4·25-s − 5.26·37-s + 2.43·43-s − 1/2·64-s + 0.977·67-s − 5.85·79-s + 4·100-s − 3.06·109-s + 2.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 5.26·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2.15·169-s − 2.43·172-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{12} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(199281.\)
Root analytic conductor: \(4.59656\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2646} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{12} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.05841748553\)
\(L(\frac12)\) \(\approx\) \(0.05841748553\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
good5$C_2$ \( ( 1 + p T^{2} )^{4} \)
11$C_2^2$ \( ( 1 - 13 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \)
17$C_2$ \( ( 1 + p T^{2} )^{4} \)
19$C_2$ \( ( 1 - p T^{2} )^{4} \)
23$C_2$ \( ( 1 - p T^{2} )^{4} \)
29$C_2^2$ \( ( 1 + 23 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2$ \( ( 1 - 11 T + p T^{2} )^{2}( 1 + 11 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 8 T + p T^{2} )^{4} \)
41$C_2^2$ \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{4} \)
47$C_2^2$ \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + 91 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \)
67$C_2$ \( ( 1 - 2 T + p T^{2} )^{4} \)
71$C_2^2$ \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 - 119 T^{2} + p^{2} T^{4} )^{2} \)
79$C_2$ \( ( 1 + 13 T + p T^{2} )^{4} \)
83$C_2^2$ \( ( 1 + 139 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 - 119 T^{2} + p^{2} T^{4} )^{2} \)
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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

−6.11027004683910583175952793596, −6.04172122608279827931198105615, −5.93547577587174152902369081047, −5.54243244906231032844768619512, −5.46566330435062002101903425304, −5.40148200973004817018802523135, −5.00795350201739234840604597492, −4.94986080820336098380206100466, −4.45494770468951284887059641161, −4.38317418358235772484216556305, −4.29607892244510893422977292974, −3.73850560760247235730777413811, −3.73739690963575306371655323228, −3.67104615046743863423325712964, −3.53444700466664810478895880049, −3.08110597321353337030747638761, −2.66734810818300502285849989552, −2.56096117314947496579523076061, −2.30404124844821287007999738350, −1.84395497427287638255664399873, −1.63367412647309236589960454103, −1.51021356338277789901656331606, −1.17415559689160526998408073769, −0.43902315514239629342500150671, −0.06293047766237565424761538857, 0.06293047766237565424761538857, 0.43902315514239629342500150671, 1.17415559689160526998408073769, 1.51021356338277789901656331606, 1.63367412647309236589960454103, 1.84395497427287638255664399873, 2.30404124844821287007999738350, 2.56096117314947496579523076061, 2.66734810818300502285849989552, 3.08110597321353337030747638761, 3.53444700466664810478895880049, 3.67104615046743863423325712964, 3.73739690963575306371655323228, 3.73850560760247235730777413811, 4.29607892244510893422977292974, 4.38317418358235772484216556305, 4.45494770468951284887059641161, 4.94986080820336098380206100466, 5.00795350201739234840604597492, 5.40148200973004817018802523135, 5.46566330435062002101903425304, 5.54243244906231032844768619512, 5.93547577587174152902369081047, 6.04172122608279827931198105615, 6.11027004683910583175952793596

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