Properties

Label 8-300e4-1.1-c2e4-0-7
Degree $8$
Conductor $8100000000$
Sign $1$
Analytic cond. $4465.03$
Root an. cond. $2.85909$
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  + 4·4-s + 6·9-s + 104·29-s + 24·36-s + 232·41-s − 100·49-s + 104·61-s − 64·64-s + 27·81-s − 328·89-s − 296·101-s + 184·109-s + 416·116-s + 388·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 928·164-s + 167-s + 668·169-s + 173-s + 179-s + ⋯
L(s)  = 1  + 4-s + 2/3·9-s + 3.58·29-s + 2/3·36-s + 5.65·41-s − 2.04·49-s + 1.70·61-s − 64-s + 1/3·81-s − 3.68·89-s − 2.93·101-s + 1.68·109-s + 3.58·116-s + 3.20·121-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 + 5.65·164-s + 0.00598·167-s + 3.95·169-s + 0.00578·173-s + 0.00558·179-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{8} \cdot 3^{4} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(4465.03\)
Root analytic conductor: \(2.85909\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{300} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{8} \cdot 3^{4} \cdot 5^{8} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(6.057126247\)
\(L(\frac12)\) \(\approx\) \(6.057126247\)
\(L(2)\) 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$C_2^2$ \( 1 - p^{2} T^{2} + p^{4} T^{4} \)
3$C_2$ \( ( 1 - p T^{2} )^{2} \)
5 \( 1 \)
good7$C_2^2$ \( ( 1 + 50 T^{2} + p^{4} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - 194 T^{2} + p^{4} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - 334 T^{2} + p^{4} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 - 478 T^{2} + p^{4} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 - 290 T^{2} + p^{4} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 + 290 T^{2} + p^{4} T^{4} )^{2} \)
29$C_2$ \( ( 1 - 26 T + p^{2} T^{2} )^{4} \)
31$C_2^2$ \( ( 1 - 1874 T^{2} + p^{4} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 - 2062 T^{2} + p^{4} T^{4} )^{2} \)
41$C_2$ \( ( 1 - 58 T + p^{2} T^{2} )^{4} \)
43$C_2^2$ \( ( 1 + 1346 T^{2} + p^{4} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 - 382 T^{2} + p^{4} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - 142 T^{2} + p^{4} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + 1150 T^{2} + p^{4} T^{4} )^{2} \)
61$C_2$ \( ( 1 - 26 T + p^{2} T^{2} )^{4} \)
67$C_2^2$ \( ( 1 + 8930 T^{2} + p^{4} T^{4} )^{2} \)
71$C_1$$\times$$C_1$ \( ( 1 - p T )^{4}( 1 + p T )^{4} \)
73$C_2^2$ \( ( 1 - 8542 T^{2} + p^{4} T^{4} )^{2} \)
79$C_2^2$ \( ( 1 + 1390 T^{2} + p^{4} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 + 11426 T^{2} + p^{4} T^{4} )^{2} \)
89$C_2$ \( ( 1 + 82 T + p^{2} T^{2} )^{4} \)
97$C_2^2$ \( ( 1 - 18814 T^{2} + p^{4} T^{4} )^{2} \)
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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

−8.265643624782410565805602507192, −8.023619955106863678175656637872, −7.79944600821344634124289252830, −7.46814456218183493527974991868, −7.35630990999148080948590064674, −6.83844828745120090773025301813, −6.72355855467881294539626396495, −6.68049722837820779770038869804, −6.30656011217088864318490102768, −5.82590736967970447174524390888, −5.78846829533214390619808945907, −5.59487103176334962874095512518, −4.95709854892453157190780095567, −4.63868614555646558522736893729, −4.50967753413399769410181074881, −4.18673373290333875589633645639, −4.01000301823539361850927910742, −3.42802634189928113137753482572, −2.83176954909788888508846346584, −2.74253590983432627857121751044, −2.64476146287557353020140330191, −1.99775882615400387046723522220, −1.55079743004161313379367721290, −0.948206505104185409518595159602, −0.68259592614840950207321827264, 0.68259592614840950207321827264, 0.948206505104185409518595159602, 1.55079743004161313379367721290, 1.99775882615400387046723522220, 2.64476146287557353020140330191, 2.74253590983432627857121751044, 2.83176954909788888508846346584, 3.42802634189928113137753482572, 4.01000301823539361850927910742, 4.18673373290333875589633645639, 4.50967753413399769410181074881, 4.63868614555646558522736893729, 4.95709854892453157190780095567, 5.59487103176334962874095512518, 5.78846829533214390619808945907, 5.82590736967970447174524390888, 6.30656011217088864318490102768, 6.68049722837820779770038869804, 6.72355855467881294539626396495, 6.83844828745120090773025301813, 7.35630990999148080948590064674, 7.46814456218183493527974991868, 7.79944600821344634124289252830, 8.023619955106863678175656637872, 8.265643624782410565805602507192

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