Properties

Label 8-1232e4-1.1-c2e4-0-0
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

\(L(\frac{3}{2})\) \(\approx\) \(0.7821588021\)
\(L(\frac12)\) \(\approx\) \(0.7821588021\)
\(L(2)\) 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

−6.74548712673059782735823056768, −6.45633757927332486480338454369, −6.32192413249289585298779440304, −6.25537441630081195133523786954, −5.91772654043961330488388394215, −5.51651462152505430860787463868, −5.44392088276792494440854671487, −5.14120241212597921199381859200, −5.12951678516407708420992873905, −4.86801870634779303582481746827, −4.44380132891713258463472375779, −4.21597043107844160179694019987, −3.89955662496109751248162859781, −3.51157028426922835478801756720, −3.18852503001976803303757302774, −3.17385597132120420944542159349, −3.02590411113083116099458247001, −2.58368150229558019645112819919, −2.55035467324904014736429444168, −2.47013344894499896326384720482, −1.93992981350260456977573418676, −1.18068152680698404969880027701, −0.69987523740431655165079512414, −0.50927860443159417301888213036, −0.24680139826550981414275168042, 0.24680139826550981414275168042, 0.50927860443159417301888213036, 0.69987523740431655165079512414, 1.18068152680698404969880027701, 1.93992981350260456977573418676, 2.47013344894499896326384720482, 2.55035467324904014736429444168, 2.58368150229558019645112819919, 3.02590411113083116099458247001, 3.17385597132120420944542159349, 3.18852503001976803303757302774, 3.51157028426922835478801756720, 3.89955662496109751248162859781, 4.21597043107844160179694019987, 4.44380132891713258463472375779, 4.86801870634779303582481746827, 5.12951678516407708420992873905, 5.14120241212597921199381859200, 5.44392088276792494440854671487, 5.51651462152505430860787463868, 5.91772654043961330488388394215, 6.25537441630081195133523786954, 6.32192413249289585298779440304, 6.45633757927332486480338454369, 6.74548712673059782735823056768

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