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Properties

Label 8-98e4-1.1-c4e4-0-0
Degree $8$
Conductor $92236816$
Sign $1$
Analytic cond. $10531.2$
Root an. cond. $3.18280$
Motivic weight $4$
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  + 16·4-s + 64·9-s + 24·11-s + 192·16-s + 104·23-s + 1.72e3·25-s − 1.40e3·29-s + 1.02e3·36-s − 3.39e3·37-s − 2.02e3·43-s + 384·44-s − 1.66e4·53-s + 2.04e3·64-s − 2.08e4·67-s − 1.98e3·71-s − 2.96e4·79-s + 5.79e3·81-s + 1.66e3·92-s + 1.53e3·99-s + 2.75e4·100-s − 4.07e3·107-s + 7.04e4·109-s + 7.87e4·113-s − 2.25e4·116-s − 1.57e4·121-s + 127-s + 131-s + ⋯
L(s)  = 1  + 4-s + 0.790·9-s + 0.198·11-s + 3/4·16-s + 0.196·23-s + 2.75·25-s − 1.67·29-s + 0.790·36-s − 2.47·37-s − 1.09·43-s + 0.198·44-s − 5.93·53-s + 1/2·64-s − 4.63·67-s − 0.393·71-s − 4.74·79-s + 0.882·81-s + 0.196·92-s + 0.156·99-s + 2.75·100-s − 0.355·107-s + 5.92·109-s + 6.16·113-s − 1.67·116-s − 1.07·121-s + 6.20e−5·127-s + 5.82e−5·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 92236816 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92236816 ^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(92236816\)    =    \(2^{4} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(10531.2\)
Root analytic conductor: \(3.18280\)
Motivic weight: \(4\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 92236816,\ (\ :2, 2, 2, 2),\ 1)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.638522178\)
\(L(\frac12)\) \(\approx\) \(1.638522178\)
\(L(3)\) 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$C_2$ \( ( 1 - p^{3} T^{2} )^{2} \)
7 \( 1 \)
good3$C_2^2:C_4$ \( 1 - 64 T^{2} - 1696 T^{4} - 64 p^{8} T^{6} + p^{16} T^{8} \)
5$C_2^2:C_4$ \( 1 - 1724 T^{2} + 1374142 T^{4} - 1724 p^{8} T^{6} + p^{16} T^{8} \)
11$D_{4}$ \( ( 1 - 12 T + 8100 T^{2} - 12 p^{4} T^{3} + p^{8} T^{4} )^{2} \)
13$C_2^2:C_4$ \( 1 - 35804 T^{2} + 1120765054 T^{4} - 35804 p^{8} T^{6} + p^{16} T^{8} \)
17$C_2^2:C_4$ \( 1 + 109328 T^{2} + 11607734400 T^{4} + 109328 p^{8} T^{6} + p^{16} T^{8} \)
19$C_2^2:C_4$ \( 1 - 349488 T^{2} + 60105434976 T^{4} - 349488 p^{8} T^{6} + p^{16} T^{8} \)
23$D_{4}$ \( ( 1 - 52 T + 518886 T^{2} - 52 p^{4} T^{3} + p^{8} T^{4} )^{2} \)
29$D_{4}$ \( ( 1 + 704 T + 1537498 T^{2} + 704 p^{4} T^{3} + p^{8} T^{4} )^{2} \)
31$C_2^2:C_4$ \( 1 - 1723364 T^{2} + 1763753061958 T^{4} - 1723364 p^{8} T^{6} + p^{16} T^{8} \)
37$D_{4}$ \( ( 1 + 1696 T + 4464834 T^{2} + 1696 p^{4} T^{3} + p^{8} T^{4} )^{2} \)
41$C_2^2:C_4$ \( 1 - 10204640 T^{2} + 41989966346560 T^{4} - 10204640 p^{8} T^{6} + p^{16} T^{8} \)
43$D_{4}$ \( ( 1 + 1012 T + 4332388 T^{2} + 1012 p^{4} T^{3} + p^{8} T^{4} )^{2} \)
47$C_2^2:C_4$ \( 1 + 523468 T^{2} - 21953751403130 T^{4} + 523468 p^{8} T^{6} + p^{16} T^{8} \)
53$D_{4}$ \( ( 1 + 8340 T + 32172990 T^{2} + 8340 p^{4} T^{3} + p^{8} T^{4} )^{2} \)
59$C_2^2:C_4$ \( 1 - 9053280 T^{2} + 308187661511040 T^{4} - 9053280 p^{8} T^{6} + p^{16} T^{8} \)
61$C_2^2:C_4$ \( 1 - 45572348 T^{2} + 900315190880446 T^{4} - 45572348 p^{8} T^{6} + p^{16} T^{8} \)
67$D_{4}$ \( ( 1 + 10408 T + 66148266 T^{2} + 10408 p^{4} T^{3} + p^{8} T^{4} )^{2} \)
71$D_{4}$ \( ( 1 + 992 T + 47974306 T^{2} + 992 p^{4} T^{3} + p^{8} T^{4} )^{2} \)
73$C_2^2:C_4$ \( 1 - 75977280 T^{2} + 3047966707757760 T^{4} - 75977280 p^{8} T^{6} + p^{16} T^{8} \)
79$D_{4}$ \( ( 1 + 14808 T + 122413578 T^{2} + 14808 p^{4} T^{3} + p^{8} T^{4} )^{2} \)
83$C_2^2:C_4$ \( 1 - 95940864 T^{2} + 6613088999857824 T^{4} - 95940864 p^{8} T^{6} + p^{16} T^{8} \)
89$C_2^2:C_4$ \( 1 - 143773584 T^{2} + 11740110102560544 T^{4} - 143773584 p^{8} T^{6} + p^{16} T^{8} \)
97$C_2^2:C_4$ \( 1 - 240657696 T^{2} + 28374663059580288 T^{4} - 240657696 p^{8} T^{6} + p^{16} T^{8} \)
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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

−9.606054699816702805520854115253, −9.069151759505528984305007828844, −8.917481187059213872296197122585, −8.666098285810390041820364983721, −8.598581131374087411535289319703, −7.78847365964386137030026195798, −7.68059906483042590801752946271, −7.28524624367948847931163293005, −7.22440140434609308480334313288, −6.88342580229423453629458982456, −6.39914027838077466608269808699, −6.19708439058273027815872030910, −5.96654984231932395010086976932, −5.48041491906414501546398360681, −4.74174624086382799585636322581, −4.72101953687322744767948293647, −4.71415736044656992790782169838, −3.74233427024350712239302019594, −3.35609601312238948972969597268, −3.00676247711538200237229246945, −2.88266363079974735994877837361, −1.69652706475250238683626984229, −1.68887264766193480416881110644, −1.38005779904334613316057255077, −0.24786432842720886198987614463, 0.24786432842720886198987614463, 1.38005779904334613316057255077, 1.68887264766193480416881110644, 1.69652706475250238683626984229, 2.88266363079974735994877837361, 3.00676247711538200237229246945, 3.35609601312238948972969597268, 3.74233427024350712239302019594, 4.71415736044656992790782169838, 4.72101953687322744767948293647, 4.74174624086382799585636322581, 5.48041491906414501546398360681, 5.96654984231932395010086976932, 6.19708439058273027815872030910, 6.39914027838077466608269808699, 6.88342580229423453629458982456, 7.22440140434609308480334313288, 7.28524624367948847931163293005, 7.68059906483042590801752946271, 7.78847365964386137030026195798, 8.598581131374087411535289319703, 8.666098285810390041820364983721, 8.917481187059213872296197122585, 9.069151759505528984305007828844, 9.606054699816702805520854115253

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