Properties

Label 8-570e4-1.1-c1e4-0-4
Degree $8$
Conductor $105560010000$
Sign $1$
Analytic cond. $429.148$
Root an. cond. $2.13341$
Motivic weight $1$
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  − 2·4-s + 2·9-s + 3·16-s + 4·19-s + 8·25-s − 24·29-s − 4·36-s + 24·41-s + 28·49-s − 24·59-s + 40·61-s − 4·64-s + 48·71-s − 8·76-s − 5·81-s − 48·89-s − 16·100-s + 48·116-s + 28·121-s + 127-s + 131-s + 137-s + 139-s + 6·144-s + 149-s + 151-s + 157-s + ⋯
L(s)  = 1  − 4-s + 2/3·9-s + 3/4·16-s + 0.917·19-s + 8/5·25-s − 4.45·29-s − 2/3·36-s + 3.74·41-s + 4·49-s − 3.12·59-s + 5.12·61-s − 1/2·64-s + 5.69·71-s − 0.917·76-s − 5/9·81-s − 5.08·89-s − 8/5·100-s + 4.45·116-s + 2.54·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1/2·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 19^{4}\)
Sign: $1$
Analytic conductor: \(429.148\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{570} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 19^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.173652342\)
\(L(\frac12)\) \(\approx\) \(2.173652342\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 + T^{2} )^{2} \)
3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
5$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
good7$C_2$ \( ( 1 - p T^{2} )^{4} \)
11$C_2$ \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \)
13$C_2^2$ \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{2} \)
17$C_2$ \( ( 1 + p T^{2} )^{4} \)
23$C_2^2$ \( ( 1 + 28 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{4} \)
31$C_2^2$ \( ( 1 - 44 T^{2} + p^{2} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 + 56 T^{2} + p^{2} T^{4} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{4} \)
43$C_2^2$ \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 - 68 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \)
59$C_2$ \( ( 1 + 6 T + p T^{2} )^{4} \)
61$C_2$ \( ( 1 - 10 T + p T^{2} )^{4} \)
67$C_2$ \( ( 1 + p T^{2} )^{4} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{4} \)
73$C_2$ \( ( 1 - 16 T + p T^{2} )^{2}( 1 + 16 T + p T^{2} )^{2} \)
79$C_2^2$ \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \)
83$C_2$ \( ( 1 + p T^{2} )^{4} \)
89$C_2$ \( ( 1 + 12 T + p T^{2} )^{4} \)
97$C_2$ \( ( 1 + p T^{2} )^{4} \)
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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

−7.68889498634096737755418216616, −7.65220600239859273594896480910, −7.09320944726372468239357634226, −7.06533608656079239618441658940, −6.90178140496929015553817338434, −6.89257716174919750285198260916, −6.13155290919767784587210879393, −5.94970730823940545140633868054, −5.55709104935704091131346245771, −5.52106008817624816236439520742, −5.46890835657423083265619613412, −5.22646297852615076052347844004, −4.58663052341913128737556087840, −4.42887746046340257859912100042, −4.24980664160823953742159140562, −3.88154751266298920017493264419, −3.78894507740846766411173293716, −3.35421839489797830890578378539, −3.27390827507081072546679474255, −2.38062397866808396091169083585, −2.31354768362619711898124154081, −2.26944524551373368276288639410, −1.31400295608587163549477759035, −1.03025907179804271530537988598, −0.57445133394908193928306761427, 0.57445133394908193928306761427, 1.03025907179804271530537988598, 1.31400295608587163549477759035, 2.26944524551373368276288639410, 2.31354768362619711898124154081, 2.38062397866808396091169083585, 3.27390827507081072546679474255, 3.35421839489797830890578378539, 3.78894507740846766411173293716, 3.88154751266298920017493264419, 4.24980664160823953742159140562, 4.42887746046340257859912100042, 4.58663052341913128737556087840, 5.22646297852615076052347844004, 5.46890835657423083265619613412, 5.52106008817624816236439520742, 5.55709104935704091131346245771, 5.94970730823940545140633868054, 6.13155290919767784587210879393, 6.89257716174919750285198260916, 6.90178140496929015553817338434, 7.06533608656079239618441658940, 7.09320944726372468239357634226, 7.65220600239859273594896480910, 7.68889498634096737755418216616

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