Properties

Label 8-1232e4-1.1-c2e4-0-1
Degree $8$
Conductor $2.304\times 10^{12}$
Sign $1$
Analytic cond. $1.26993\times 10^{6}$
Root an. cond. $5.79392$
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  + 28·7-s + 44·11-s + 100·25-s + 490·49-s + 1.23e3·77-s + 146·81-s + 152·107-s + 328·113-s + 1.21e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 2.80e3·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  + 4·7-s + 4·11-s + 4·25-s + 10·49-s + 16·77-s + 1.80·81-s + 1.42·107-s + 2.90·113-s + 10·121-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 + 0.00578·173-s + 16·175-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

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 7^{4} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(1.26993\times 10^{6}\)
Root analytic conductor: \(5.79392\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1232} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 7^{4} \cdot 11^{4} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(33.04916187\)
\(L(\frac12)\) \(\approx\) \(33.04916187\)
\(L(2)\) 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 \( 1 \)
7$C_1$ \( ( 1 - p T )^{4} \)
11$C_1$ \( ( 1 - p T )^{4} \)
good3$C_2^3$ \( 1 - 146 T^{4} + p^{8} T^{8} \)
5$C_1$$\times$$C_1$ \( ( 1 - p T )^{4}( 1 + p T )^{4} \)
13$C_2^3$ \( 1 - 12178 T^{4} + p^{8} T^{8} \)
17$C_2^3$ \( 1 + 142094 T^{4} + p^{8} T^{8} \)
19$C_1$$\times$$C_1$ \( ( 1 - p T )^{4}( 1 + p T )^{4} \)
23$C_1$$\times$$C_1$ \( ( 1 - p T )^{4}( 1 + p T )^{4} \)
29$C_1$$\times$$C_1$ \( ( 1 - p T )^{4}( 1 + p T )^{4} \)
31$C_2^3$ \( 1 + 1378574 T^{4} + p^{8} T^{8} \)
37$C_2^2$ \( ( 1 - 34 T^{2} + p^{4} T^{4} )^{2} \)
41$C_2^3$ \( 1 + 2255822 T^{4} + p^{8} T^{8} \)
43$C_2^2$ \( ( 1 + 926 T^{2} + p^{4} T^{4} )^{2} \)
47$C_2^3$ \( 1 - 6675826 T^{4} + p^{8} T^{8} \)
53$C_2^2$ \( ( 1 + 2846 T^{2} + p^{4} T^{4} )^{2} \)
59$C_2^3$ \( 1 - 13711186 T^{4} + p^{8} T^{8} \)
61$C_2^3$ \( 1 + 12099182 T^{4} + p^{8} T^{8} \)
67$C_1$$\times$$C_1$ \( ( 1 - p T )^{4}( 1 + p T )^{4} \)
71$C_1$$\times$$C_1$ \( ( 1 - p T )^{4}( 1 + p T )^{4} \)
73$C_2^3$ \( 1 + 56328014 T^{4} + p^{8} T^{8} \)
79$C_2^2$ \( ( 1 - 12466 T^{2} + p^{4} T^{4} )^{2} \)
83$C_1$$\times$$C_1$ \( ( 1 - p T )^{4}( 1 + p T )^{4} \)
89$C_1$$\times$$C_1$ \( ( 1 - p T )^{4}( 1 + p T )^{4} \)
97$C_1$$\times$$C_1$ \( ( 1 - p T )^{4}( 1 + p T )^{4} \)
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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

−6.90322219326583197961280507722, −6.63076090754721892186070016631, −6.18023196809465826248240270118, −6.16839138729592271747687604273, −5.93348480965602259866691937247, −5.47837946577533208722660749869, −5.36618704009420138766535012429, −5.14068108765951700144920463605, −4.77058466920446620088556807146, −4.57194882778879165985607180782, −4.52433740848850384844306117231, −4.32073031884915755640202991835, −4.30906182610289404629670747156, −3.74008427678792487615628364728, −3.46372039449342272402888365146, −3.20289741964265452223069444323, −3.18939090163165474189233959036, −2.27580770174194441142870712383, −2.23519349330700374696251749283, −2.04405767608905749476009060661, −1.55799519577792651295570905654, −1.42599417821915204199863726690, −0.923437320059324116796012468151, −0.907745116996535980540513052365, −0.887831349199614192755749214926, 0.887831349199614192755749214926, 0.907745116996535980540513052365, 0.923437320059324116796012468151, 1.42599417821915204199863726690, 1.55799519577792651295570905654, 2.04405767608905749476009060661, 2.23519349330700374696251749283, 2.27580770174194441142870712383, 3.18939090163165474189233959036, 3.20289741964265452223069444323, 3.46372039449342272402888365146, 3.74008427678792487615628364728, 4.30906182610289404629670747156, 4.32073031884915755640202991835, 4.52433740848850384844306117231, 4.57194882778879165985607180782, 4.77058466920446620088556807146, 5.14068108765951700144920463605, 5.36618704009420138766535012429, 5.47837946577533208722660749869, 5.93348480965602259866691937247, 6.16839138729592271747687604273, 6.18023196809465826248240270118, 6.63076090754721892186070016631, 6.90322219326583197961280507722

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