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Properties

Label 8-75e4-1.1-c11e4-0-3
Degree $8$
Conductor $31640625$
Sign $1$
Analytic cond. $1.10272\times 10^{7}$
Root an. cond. $7.59116$
Motivic weight $11$
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  + 6.28e6·16-s + 7.28e8·31-s − 5.21e10·61-s − 3.13e10·81-s + 1.14e12·121-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 + 229-s + 233-s + 239-s + ⋯
L(s)  = 1  + 1.49·16-s + 4.56·31-s − 7.89·61-s − 81-s + 4·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s+11/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(31640625\)    =    \(3^{4} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(1.10272\times 10^{7}\)
Root analytic conductor: \(7.59116\)
Motivic weight: \(11\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 31640625,\ (\ :11/2, 11/2, 11/2, 11/2),\ 1)\)

Particular Values

\(L(6)\) \(\approx\) \(3.783900652\)
\(L(\frac12)\) \(\approx\) \(3.783900652\)
\(L(\frac{13}{2})\) 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)$
bad3$C_2^2$ \( 1 + p^{22} T^{4} \)
5 \( 1 \)
good2$C_2^3$ \( 1 - 6283207 T^{4} + p^{44} T^{8} \)
7$C_2^2$ \( ( 1 + p^{22} T^{4} )^{2} \)
11$C_2$ \( ( 1 - p^{11} T^{2} )^{4} \)
13$C_2^2$ \( ( 1 + p^{22} T^{4} )^{2} \)
17$C_2^3$ \( 1 + \)\(23\!\cdots\!18\)\( T^{4} + p^{44} T^{8} \)
19$C_2^2$ \( ( 1 + 113455657141178 T^{2} + p^{22} T^{4} )^{2} \)
23$C_2^3$ \( 1 - \)\(15\!\cdots\!02\)\( T^{4} + p^{44} T^{8} \)
29$C_2$ \( ( 1 + p^{11} T^{2} )^{4} \)
31$C_2$ \( ( 1 - 182093992 T + p^{11} T^{2} )^{4} \)
37$C_2^2$ \( ( 1 + p^{22} T^{4} )^{2} \)
41$C_2$ \( ( 1 - p^{11} T^{2} )^{4} \)
43$C_2^2$ \( ( 1 + p^{22} T^{4} )^{2} \)
47$C_2^3$ \( 1 + \)\(12\!\cdots\!78\)\( T^{4} + p^{44} T^{8} \)
53$C_2^3$ \( 1 + \)\(67\!\cdots\!78\)\( T^{4} + p^{44} T^{8} \)
59$C_2$ \( ( 1 + p^{11} T^{2} )^{4} \)
61$C_2$ \( ( 1 + 13027614598 T + p^{11} T^{2} )^{4} \)
67$C_2^2$ \( ( 1 + p^{22} T^{4} )^{2} \)
71$C_2$ \( ( 1 - p^{11} T^{2} )^{4} \)
73$C_2^2$ \( ( 1 + p^{22} T^{4} )^{2} \)
79$C_2^2$ \( ( 1 - \)\(13\!\cdots\!02\)\( T^{2} + p^{22} T^{4} )^{2} \)
83$C_2^3$ \( 1 - \)\(17\!\cdots\!62\)\( T^{4} + p^{44} T^{8} \)
89$C_2$ \( ( 1 + p^{11} T^{2} )^{4} \)
97$C_2^2$ \( ( 1 + p^{22} T^{4} )^{2} \)
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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

−8.425983557439146311175068790812, −7.989869343767337632982218438190, −7.74999413035725731062181163912, −7.74195374561750612036775540642, −7.41750054778320087866929222039, −6.70783132345898204698487870229, −6.55533323458831590557073698943, −6.24584839116450058780282009740, −6.03470099642889825418689293518, −5.69662418029014465936442051184, −5.35805727848969243888320592275, −4.73151460924566387877939966660, −4.59937085120024305501239932527, −4.40161138104076099949549503650, −4.10480998970486169161019890882, −3.25231205176318709536097357560, −3.09076792260986252457063774544, −3.08601309639529090857987222632, −2.63380082580323202483329053876, −2.06219747113037018415044503624, −1.59322162548126943097496250474, −1.32200071426491158921396991772, −1.04032624956134079912957134477, −0.61944599774270272998193452695, −0.24972117325237790715009141800, 0.24972117325237790715009141800, 0.61944599774270272998193452695, 1.04032624956134079912957134477, 1.32200071426491158921396991772, 1.59322162548126943097496250474, 2.06219747113037018415044503624, 2.63380082580323202483329053876, 3.08601309639529090857987222632, 3.09076792260986252457063774544, 3.25231205176318709536097357560, 4.10480998970486169161019890882, 4.40161138104076099949549503650, 4.59937085120024305501239932527, 4.73151460924566387877939966660, 5.35805727848969243888320592275, 5.69662418029014465936442051184, 6.03470099642889825418689293518, 6.24584839116450058780282009740, 6.55533323458831590557073698943, 6.70783132345898204698487870229, 7.41750054778320087866929222039, 7.74195374561750612036775540642, 7.74999413035725731062181163912, 7.989869343767337632982218438190, 8.425983557439146311175068790812

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