Properties

Label 8-2352e4-1.1-c1e4-0-2
Degree $8$
Conductor $3.060\times 10^{13}$
Sign $1$
Analytic cond. $124410.$
Root an. cond. $4.33368$
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  − 6·9-s − 8·25-s − 32·37-s + 27·81-s − 40·109-s + 40·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 52·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 48·225-s + 227-s + ⋯
L(s)  = 1  − 2·9-s − 8/5·25-s − 5.26·37-s + 3·81-s − 3.83·109-s + 3.63·121-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 − 4·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 16/5·225-s + 0.0663·227-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 3^{4} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(124410.\)
Root analytic conductor: \(4.33368\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2352} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 3^{4} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.1553570629\)
\(L(\frac12)\) \(\approx\) \(0.1553570629\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_2$ \( ( 1 + p T^{2} )^{2} \)
7 \( 1 \)
good5$C_2^2$ \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2$ \( ( 1 + p T^{2} )^{4} \)
17$C_2^2$ \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2$ \( ( 1 - p T^{2} )^{4} \)
31$C_2^2$ \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \)
37$C_2$ \( ( 1 + 8 T + p T^{2} )^{4} \)
41$C_2^2$ \( ( 1 - 68 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2$ \( ( 1 - p T^{2} )^{4} \)
47$C_2$ \( ( 1 + p T^{2} )^{4} \)
53$C_2$ \( ( 1 - p T^{2} )^{4} \)
59$C_2$ \( ( 1 + p T^{2} )^{4} \)
61$C_2$ \( ( 1 + p T^{2} )^{4} \)
67$C_2$ \( ( 1 - p T^{2} )^{4} \)
71$C_2^2$ \( ( 1 + 100 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2$ \( ( 1 + p T^{2} )^{4} \)
79$C_2$ \( ( 1 - p T^{2} )^{4} \)
83$C_2$ \( ( 1 + p T^{2} )^{4} \)
89$C_2^2$ \( ( 1 + 172 T^{2} + p^{2} T^{4} )^{2} \)
97$C_2$ \( ( 1 + p T^{2} )^{4} \)
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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

−6.29706260960831750613270686979, −6.29291272895179281096972381301, −5.80898808755071121168295168030, −5.75108511108952738878256909680, −5.45339467614424216689196611988, −5.41961881032805411772378256416, −5.31621290832368697493696847915, −4.94892227586108203402590538798, −4.70131636481516264745009897332, −4.61877063343230696467379275239, −4.23988108097831991687896551062, −3.85710675981420994398508145602, −3.71782093766464687673275235877, −3.57551004297520296104969769845, −3.27696543770850333444925522531, −3.21533409493816358890282660618, −2.91716605522177245047374548203, −2.45038794539872617361860811970, −2.23624399118433364632000541698, −2.22475343223106269201023993982, −1.70430002482312423843975471442, −1.52049323604811346340513537758, −1.18272912914309086878057315245, −0.48232553811209032471025234907, −0.096655376359695728580090449323, 0.096655376359695728580090449323, 0.48232553811209032471025234907, 1.18272912914309086878057315245, 1.52049323604811346340513537758, 1.70430002482312423843975471442, 2.22475343223106269201023993982, 2.23624399118433364632000541698, 2.45038794539872617361860811970, 2.91716605522177245047374548203, 3.21533409493816358890282660618, 3.27696543770850333444925522531, 3.57551004297520296104969769845, 3.71782093766464687673275235877, 3.85710675981420994398508145602, 4.23988108097831991687896551062, 4.61877063343230696467379275239, 4.70131636481516264745009897332, 4.94892227586108203402590538798, 5.31621290832368697493696847915, 5.41961881032805411772378256416, 5.45339467614424216689196611988, 5.75108511108952738878256909680, 5.80898808755071121168295168030, 6.29291272895179281096972381301, 6.29706260960831750613270686979

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