Properties

Label 8-960e4-1.1-c2e4-0-2
Degree $8$
Conductor $849346560000$
Sign $1$
Analytic cond. $468193.$
Root an. cond. $5.11449$
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  + 2·9-s − 24·19-s − 30·25-s − 136·31-s + 68·49-s − 296·61-s + 312·79-s − 77·81-s − 296·109-s + 324·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 388·169-s − 48·171-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯
L(s)  = 1  + 2/9·9-s − 1.26·19-s − 6/5·25-s − 4.38·31-s + 1.38·49-s − 4.85·61-s + 3.94·79-s − 0.950·81-s − 2.71·109-s + 2.67·121-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 + 2.29·169-s − 0.280·171-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{24} \cdot 3^{4} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(468193.\)
Root analytic conductor: \(5.11449\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{24} \cdot 3^{4} \cdot 5^{4} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.3307382738\)
\(L(\frac12)\) \(\approx\) \(0.3307382738\)
\(L(2)\) 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 \( 1 \)
3$C_2^2$ \( 1 - 2 T^{2} + p^{4} T^{4} \)
5$C_2^2$ \( 1 + 6 p T^{2} + p^{4} T^{4} \)
good7$C_2^2$ \( ( 1 - 34 T^{2} + p^{4} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - 162 T^{2} + p^{4} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - 194 T^{2} + p^{4} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 - 402 T^{2} + p^{4} T^{4} )^{2} \)
19$C_2$ \( ( 1 + 6 T + p^{2} T^{2} )^{4} \)
23$C_2^2$ \( ( 1 + 1038 T^{2} + p^{4} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 962 T^{2} + p^{4} T^{4} )^{2} \)
31$C_2$ \( ( 1 + 34 T + p^{2} T^{2} )^{4} \)
37$C_2^2$ \( ( 1 - 802 T^{2} + p^{4} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 - 3042 T^{2} + p^{4} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 - 2914 T^{2} + p^{4} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 + 4398 T^{2} + p^{4} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 + 3998 T^{2} + p^{4} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + 2718 T^{2} + p^{4} T^{4} )^{2} \)
61$C_2$ \( ( 1 + 74 T + p^{2} T^{2} )^{4} \)
67$C_2^2$ \( ( 1 - 514 T^{2} + p^{4} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 - 7202 T^{2} + p^{4} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 - 7522 T^{2} + p^{4} T^{4} )^{2} \)
79$C_2$ \( ( 1 - 78 T + p^{2} T^{2} )^{4} \)
83$C_2^2$ \( ( 1 + 3198 T^{2} + p^{4} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 - 15522 T^{2} + p^{4} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 - 17794 T^{2} + p^{4} T^{4} )^{2} \)
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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

−7.07252229252093714449022158686, −6.93492592047820895183909839816, −6.43913582198432812563487540614, −6.21869073284153128396092482381, −6.05642587410979766359741297600, −5.95104285922947198801673972190, −5.60997791787082784396805071162, −5.37000575542671209361176081780, −5.19307022530198434198306701006, −4.91065164978837448556353049093, −4.62348672661812909043474462784, −4.38050865857605679457396053425, −4.00398209765649515506036594233, −3.92233544001999396905447563872, −3.68376029292752507421317710368, −3.37059281104143024782426337900, −3.19218917468931507695500974511, −2.59798153272257526442868259051, −2.55121825752279001289322625441, −1.99444385430759652777462243598, −1.76549268082112390424949835283, −1.71328367436028077459072079993, −1.21823706284386630488458937993, −0.53798223621897566175362019500, −0.11105871862258896519504784525, 0.11105871862258896519504784525, 0.53798223621897566175362019500, 1.21823706284386630488458937993, 1.71328367436028077459072079993, 1.76549268082112390424949835283, 1.99444385430759652777462243598, 2.55121825752279001289322625441, 2.59798153272257526442868259051, 3.19218917468931507695500974511, 3.37059281104143024782426337900, 3.68376029292752507421317710368, 3.92233544001999396905447563872, 4.00398209765649515506036594233, 4.38050865857605679457396053425, 4.62348672661812909043474462784, 4.91065164978837448556353049093, 5.19307022530198434198306701006, 5.37000575542671209361176081780, 5.60997791787082784396805071162, 5.95104285922947198801673972190, 6.05642587410979766359741297600, 6.21869073284153128396092482381, 6.43913582198432812563487540614, 6.93492592047820895183909839816, 7.07252229252093714449022158686

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