Properties

Label 8-10e8-1.1-c8e4-0-1
Degree $8$
Conductor $100000000$
Sign $1$
Analytic cond. $2.75418\times 10^{6}$
Root an. cond. $6.38262$
Motivic weight $8$
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  + 1.68e3·11-s + 2.51e6·31-s + 1.38e7·41-s + 9.71e7·61-s + 1.45e8·71-s + 4.37e7·81-s − 1.27e8·101-s − 8.55e8·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯
L(s)  = 1  + 0.114·11-s + 2.71·31-s + 4.89·41-s + 7.01·61-s + 5.71·71-s + 1.01·81-s − 1.22·101-s − 3.99·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(9-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+4)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{8} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(2.75418\times 10^{6}\)
Root analytic conductor: \(6.38262\)
Motivic weight: \(8\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{8} \cdot 5^{8} ,\ ( \ : 4, 4, 4, 4 ),\ 1 )\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(7.272392762\)
\(L(\frac12)\) \(\approx\) \(7.272392762\)
\(L(5)\) 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 \( 1 \)
5 \( 1 \)
good3$C_2^3$ \( 1 - 4865714 p^{2} T^{4} + p^{32} T^{8} \)
7$C_2^3$ \( 1 + 24611967998 p^{4} T^{4} + p^{32} T^{8} \)
11$C_2$ \( ( 1 - 420 T + p^{8} T^{2} )^{4} \)
13$C_2^3$ \( 1 + 200454714069053954 T^{4} + p^{32} T^{8} \)
17$C_2^3$ \( 1 - 81836647942062791806 T^{4} + p^{32} T^{8} \)
19$C_2^2$ \( ( 1 - 93532258 p^{2} T^{2} + p^{16} T^{4} )^{2} \)
23$C_2^3$ \( 1 - \)\(32\!\cdots\!22\)\( T^{4} + p^{32} T^{8} \)
29$C_2^2$ \( ( 1 - 937420521758 T^{2} + p^{16} T^{4} )^{2} \)
31$C_2$ \( ( 1 - 627976 T + p^{8} T^{2} )^{4} \)
37$C_2^3$ \( 1 - \)\(16\!\cdots\!82\)\( T^{4} + p^{32} T^{8} \)
41$C_2$ \( ( 1 - 3461016 T + p^{8} T^{2} )^{4} \)
43$C_2^3$ \( 1 - \)\(21\!\cdots\!98\)\( T^{4} + p^{32} T^{8} \)
47$C_2^3$ \( 1 + \)\(21\!\cdots\!74\)\( T^{4} + p^{32} T^{8} \)
53$C_2^3$ \( 1 + \)\(51\!\cdots\!58\)\( T^{4} + p^{32} T^{8} \)
59$C_2^2$ \( ( 1 - 111680667288626 T^{2} + p^{16} T^{4} )^{2} \)
61$C_2$ \( ( 1 - 24284432 T + p^{8} T^{2} )^{4} \)
67$C_2^3$ \( 1 - \)\(26\!\cdots\!62\)\( T^{4} + p^{32} T^{8} \)
71$C_2$ \( ( 1 - 36337008 T + p^{8} T^{2} )^{4} \)
73$C_2^3$ \( 1 - \)\(10\!\cdots\!86\)\( T^{4} + p^{32} T^{8} \)
79$C_2^2$ \( ( 1 - 2174687975647426 T^{2} + p^{16} T^{4} )^{2} \)
83$C_2^3$ \( 1 - \)\(80\!\cdots\!62\)\( T^{4} + p^{32} T^{8} \)
89$C_2^2$ \( ( 1 - 7819700304303038 T^{2} + p^{16} T^{4} )^{2} \)
97$C_2^3$ \( 1 - \)\(11\!\cdots\!42\)\( T^{4} + p^{32} T^{8} \)
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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

−8.527554746494703816802756249861, −8.136719483830492668842455713557, −7.961945681590457225693885109417, −7.78127429379782835530150506623, −7.63776416526739076503683341092, −6.83189020615364606737820156692, −6.67030960888784217769678152884, −6.56395080067568035436288030926, −6.37331120103137948134377595685, −5.76536379520926824725687042517, −5.30713061590918909546209171180, −5.23777886249820827884923461101, −5.06951152179453231037906655869, −4.18089950912373555953980515566, −4.15443417274065027719482283680, −3.92865559251209604583404900433, −3.57445109686916402073941030506, −2.88214826877039764902045717410, −2.38591889028467664637943897685, −2.36652565405389655473021689192, −2.26416979785946066977442742429, −1.06695379903639156842807263870, −1.00662392204109678114375776006, −0.870903627817817694143726239398, −0.37821866136852964399809706502, 0.37821866136852964399809706502, 0.870903627817817694143726239398, 1.00662392204109678114375776006, 1.06695379903639156842807263870, 2.26416979785946066977442742429, 2.36652565405389655473021689192, 2.38591889028467664637943897685, 2.88214826877039764902045717410, 3.57445109686916402073941030506, 3.92865559251209604583404900433, 4.15443417274065027719482283680, 4.18089950912373555953980515566, 5.06951152179453231037906655869, 5.23777886249820827884923461101, 5.30713061590918909546209171180, 5.76536379520926824725687042517, 6.37331120103137948134377595685, 6.56395080067568035436288030926, 6.67030960888784217769678152884, 6.83189020615364606737820156692, 7.63776416526739076503683341092, 7.78127429379782835530150506623, 7.961945681590457225693885109417, 8.136719483830492668842455713557, 8.527554746494703816802756249861

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