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Properties

Label 8-45e4-1.1-c2e4-0-0
Degree $8$
Conductor $4100625$
Sign $1$
Analytic cond. $2.26042$
Root an. cond. $1.10732$
Motivic weight $2$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 20·7-s + 40·13-s + 23·16-s − 40·25-s + 32·31-s + 40·37-s + 40·43-s + 200·49-s − 232·61-s + 280·67-s + 220·73-s − 800·91-s − 20·97-s − 140·103-s − 460·112-s + 16·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 800·169-s + 173-s + ⋯
L(s)  = 1  − 2.85·7-s + 3.07·13-s + 1.43·16-s − 8/5·25-s + 1.03·31-s + 1.08·37-s + 0.930·43-s + 4.08·49-s − 3.80·61-s + 4.17·67-s + 3.01·73-s − 8.79·91-s − 0.206·97-s − 1.35·103-s − 4.10·112-s + 0.132·121-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 + 4.73·169-s + 0.00578·173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4100625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4100625 ^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(4100625\)    =    \(3^{8} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(2.26042\)
Root analytic conductor: \(1.10732\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 4100625,\ (\ :1, 1, 1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.048905771\)
\(L(\frac12)\) \(\approx\) \(1.048905771\)
\(L(2)\) 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)$
bad3 \( 1 \)
5$C_2^2$ \( 1 + 8 p T^{2} + p^{4} T^{4} \)
good2$C_2^3$ \( 1 - 23 T^{4} + p^{8} T^{8} \)
7$C_2^2$ \( ( 1 + 10 T + 50 T^{2} + 10 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - 8 T^{2} + p^{4} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - 20 T + 200 T^{2} - 20 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
17$C_2^3$ \( 1 + 144322 T^{4} + p^{8} T^{8} \)
19$C_2^2$ \( ( 1 - 398 T^{2} + p^{4} T^{4} )^{2} \)
23$C_2^3$ \( 1 + 517762 T^{4} + p^{8} T^{8} \)
29$C_2^2$ \( ( 1 + 568 T^{2} + p^{4} T^{4} )^{2} \)
31$C_2$ \( ( 1 - 8 T + p^{2} T^{2} )^{4} \)
37$C_2^2$ \( ( 1 - 20 T + 200 T^{2} - 20 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 + 2362 T^{2} + p^{4} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 - 20 T + 200 T^{2} - 20 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
47$C_2^3$ \( 1 - 8681918 T^{4} + p^{8} T^{8} \)
53$C_2^3$ \( 1 + 3037282 T^{4} + p^{8} T^{8} \)
59$C_2^2$ \( ( 1 - 4712 T^{2} + p^{4} T^{4} )^{2} \)
61$C_2$ \( ( 1 + 58 T + p^{2} T^{2} )^{4} \)
67$C_2^2$ \( ( 1 - 140 T + 9800 T^{2} - 140 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 + 6082 T^{2} + p^{4} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 - 110 T + 6050 T^{2} - 110 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
79$C_2^2$ \( ( 1 - 12338 T^{2} + p^{4} T^{4} )^{2} \)
83$C_2^3$ \( 1 - 30948638 T^{4} + p^{8} T^{8} \)
89$C_1$$\times$$C_1$ \( ( 1 - p T )^{4}( 1 + p T )^{4} \)
97$C_2^2$ \( ( 1 + 10 T + 50 T^{2} + 10 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
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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

−11.55497479671012218091792319421, −11.25567763653390900346230388590, −10.99449609383926143649597317802, −10.70539611934926640009774025482, −10.22841955442148026022444801860, −10.09830133353748576268061770092, −9.531570061267914645515259405950, −9.460043886029300855700829233784, −9.260270254761943393254256602495, −8.715102600631252747470021527276, −8.143899248717885305180623709484, −8.107359053703639145676963085828, −7.75689356288320329873923647539, −7.01301774144573481201652635605, −6.61915701757328771252369196864, −6.34599495110069946504508343750, −6.05875941023193682378623855277, −5.85988594623789790079338906672, −5.45630548153638002365264185868, −4.47481000161295481147897288915, −3.82324094743719083889806152585, −3.46164702436307439002314720361, −3.44484010972461472724905766649, −2.52187720425739144518409042233, −1.02551862370993760166493414529, 1.02551862370993760166493414529, 2.52187720425739144518409042233, 3.44484010972461472724905766649, 3.46164702436307439002314720361, 3.82324094743719083889806152585, 4.47481000161295481147897288915, 5.45630548153638002365264185868, 5.85988594623789790079338906672, 6.05875941023193682378623855277, 6.34599495110069946504508343750, 6.61915701757328771252369196864, 7.01301774144573481201652635605, 7.75689356288320329873923647539, 8.107359053703639145676963085828, 8.143899248717885305180623709484, 8.715102600631252747470021527276, 9.260270254761943393254256602495, 9.460043886029300855700829233784, 9.531570061267914645515259405950, 10.09830133353748576268061770092, 10.22841955442148026022444801860, 10.70539611934926640009774025482, 10.99449609383926143649597317802, 11.25567763653390900346230388590, 11.55497479671012218091792319421

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