Properties

Label 8-1152e4-1.1-c2e4-0-13
Degree $8$
Conductor $1.761\times 10^{12}$
Sign $1$
Analytic cond. $970845.$
Root an. cond. $5.60265$
Motivic weight $2$
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  + 72·17-s + 68·25-s − 72·41-s + 100·49-s − 328·73-s − 504·89-s + 440·97-s + 504·113-s − 388·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 676·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  + 4.23·17-s + 2.71·25-s − 1.75·41-s + 2.04·49-s − 4.49·73-s − 5.66·89-s + 4.53·97-s + 4.46·113-s − 3.20·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 4·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{28} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(970845.\)
Root analytic conductor: \(5.60265\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{28} \cdot 3^{8} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(7.475207552\)
\(L(\frac12)\) \(\approx\) \(7.475207552\)
\(L(2)\) 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 \( 1 \)
good5$C_2^2$ \( ( 1 - 34 T^{2} + p^{4} T^{4} )^{2} \)
7$C_2^2$ \( ( 1 - 50 T^{2} + p^{4} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 + 194 T^{2} + p^{4} T^{4} )^{2} \)
13$C_1$$\times$$C_1$ \( ( 1 - p T )^{4}( 1 + p T )^{4} \)
17$C_2$ \( ( 1 - 18 T + p^{2} T^{2} )^{4} \)
19$C_2^2$ \( ( 1 + 290 T^{2} + p^{4} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 + 670 T^{2} + p^{4} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 1666 T^{2} + p^{4} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 + 430 T^{2} + p^{4} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 + 2446 T^{2} + p^{4} T^{4} )^{2} \)
41$C_2$ \( ( 1 + 18 T + p^{2} T^{2} )^{4} \)
43$C_2^2$ \( ( 1 - 190 T^{2} + p^{4} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 - 2690 T^{2} + p^{4} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - 3682 T^{2} + p^{4} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + 3074 T^{2} + p^{4} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 - 2258 T^{2} + p^{4} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 + 8546 T^{2} + p^{4} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 - 8354 T^{2} + p^{4} T^{4} )^{2} \)
73$C_2$ \( ( 1 + 82 T + p^{2} T^{2} )^{4} \)
79$C_2^2$ \( ( 1 - 8594 T^{2} + p^{4} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 - 3550 T^{2} + p^{4} T^{4} )^{2} \)
89$C_2$ \( ( 1 + 126 T + p^{2} T^{2} )^{4} \)
97$C_2$ \( ( 1 - 110 T + p^{2} T^{2} )^{4} \)
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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

−6.86422012190438359078361645025, −6.57644123522675135890136743314, −6.50491828713032080648408296047, −5.86257168586347795933530374307, −5.84985096964436186265942254322, −5.68859687183778914514713600335, −5.60964787291047453358608906415, −5.17344111271199071748647454401, −5.15432709463440302168375174477, −4.75023910016741785738829114648, −4.52605682725075321197939644386, −4.31167353552181212593726436827, −4.05892552590043108589947529837, −3.57673556573545803613706538956, −3.36012321745227300968423063325, −3.33760543740613650169015531026, −2.96244552248747365841215365970, −2.73650832318037320553074631291, −2.66505200497999106628863372588, −1.96490530206499529636561366592, −1.59950988926846922614105142926, −1.29978819046718036379502081392, −1.19489227647182569626389076944, −0.68575365698402557343219794074, −0.44067603246664855991975339588, 0.44067603246664855991975339588, 0.68575365698402557343219794074, 1.19489227647182569626389076944, 1.29978819046718036379502081392, 1.59950988926846922614105142926, 1.96490530206499529636561366592, 2.66505200497999106628863372588, 2.73650832318037320553074631291, 2.96244552248747365841215365970, 3.33760543740613650169015531026, 3.36012321745227300968423063325, 3.57673556573545803613706538956, 4.05892552590043108589947529837, 4.31167353552181212593726436827, 4.52605682725075321197939644386, 4.75023910016741785738829114648, 5.15432709463440302168375174477, 5.17344111271199071748647454401, 5.60964787291047453358608906415, 5.68859687183778914514713600335, 5.84985096964436186265942254322, 5.86257168586347795933530374307, 6.50491828713032080648408296047, 6.57644123522675135890136743314, 6.86422012190438359078361645025

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