Properties

Label 8-15e8-1.1-c3e4-0-1
Degree $8$
Conductor $2562890625$
Sign $1$
Analytic cond. $31059.4$
Root an. cond. $3.64354$
Motivic weight $3$
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  + 36·7-s − 252·13-s + 79·16-s + 784·31-s − 828·37-s + 576·43-s + 648·49-s − 1.28e3·61-s − 1.51e3·67-s − 1.51e3·73-s − 9.07e3·91-s + 1.00e3·97-s − 3.78e3·103-s + 2.84e3·112-s + 2.40e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 3.17e4·169-s + 173-s + 179-s + ⋯
L(s)  = 1  + 1.94·7-s − 5.37·13-s + 1.23·16-s + 4.54·31-s − 3.67·37-s + 2.04·43-s + 1.88·49-s − 2.70·61-s − 2.75·67-s − 2.42·73-s − 10.4·91-s + 1.05·97-s − 3.61·103-s + 2.39·112-s + 1.80·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 14.4·169-s + 0.000439·173-s + 0.000417·179-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(3^{8} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(31059.4\)
Root analytic conductor: \(3.64354\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{225} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{8} \cdot 5^{8} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(0.2911067696\)
\(L(\frac12)\) \(\approx\) \(0.2911067696\)
\(L(\frac{5}{2})\) 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 \( 1 \)
good2$C_2^3$ \( 1 - 79 T^{4} + p^{12} T^{8} \)
7$C_2^2$ \( ( 1 - 18 T + 162 T^{2} - 18 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - 1204 T^{2} + p^{6} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 + 126 T + 7938 T^{2} + 126 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
17$C_2^3$ \( 1 + 16720706 T^{4} + p^{12} T^{8} \)
19$C_2^2$ \( ( 1 - 8818 T^{2} + p^{6} T^{4} )^{2} \)
23$C_2^3$ \( 1 - 102026878 T^{4} + p^{12} T^{8} \)
29$C_2^2$ \( ( 1 - 3710 T^{2} + p^{6} T^{4} )^{2} \)
31$C_2$ \( ( 1 - 196 T + p^{3} T^{2} )^{4} \)
37$C_2^2$ \( ( 1 + 414 T + 85698 T^{2} + 414 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 - 66400 T^{2} + p^{6} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 - 288 T + 41472 T^{2} - 288 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
47$C_2^3$ \( 1 - 19408253758 T^{4} + p^{12} T^{8} \)
53$C_2^3$ \( 1 - 718 p^{4} T^{4} + p^{12} T^{8} \)
59$C_2^2$ \( ( 1 + 339316 T^{2} + p^{6} T^{4} )^{2} \)
61$C_2$ \( ( 1 + 322 T + p^{3} T^{2} )^{4} \)
67$C_2^2$ \( ( 1 + 756 T + 285768 T^{2} + 756 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 - 10150 T^{2} + p^{6} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 + 756 T + 285768 T^{2} + 756 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
79$C_2^2$ \( ( 1 - 747934 T^{2} + p^{6} T^{4} )^{2} \)
83$C_2^3$ \( 1 - 651490514638 T^{4} + p^{12} T^{8} \)
89$C_2^2$ \( ( 1 + 1338496 T^{2} + p^{6} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 - 504 T + 127008 T^{2} - 504 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
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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

−8.384196639544068837722160842646, −8.085896519930092250078765560673, −7.77534750555935756620399908332, −7.61329339867146147465158685342, −7.55993917983382946828184028512, −7.38887704653494610495339774320, −6.82890121268232047062070593990, −6.72104970897846474699859925754, −6.46362954514640221937282889832, −5.71877439685866626388071652128, −5.55879025373941386978520988296, −5.44249788431659229995224859387, −4.95743222767456540934319595443, −4.58561323294128995727647129731, −4.58347702324232167461214952940, −4.58288627526350190147756870882, −4.08021496943624331714462864893, −3.13360385758805915106042077520, −2.87739842452781200975224862272, −2.80142193979823979764278044533, −2.33251615634815889072121848626, −1.85938931847137398739171201743, −1.51403901252682886968264766980, −0.919806287717927038179160568658, −0.10547230665614115193638576772, 0.10547230665614115193638576772, 0.919806287717927038179160568658, 1.51403901252682886968264766980, 1.85938931847137398739171201743, 2.33251615634815889072121848626, 2.80142193979823979764278044533, 2.87739842452781200975224862272, 3.13360385758805915106042077520, 4.08021496943624331714462864893, 4.58288627526350190147756870882, 4.58347702324232167461214952940, 4.58561323294128995727647129731, 4.95743222767456540934319595443, 5.44249788431659229995224859387, 5.55879025373941386978520988296, 5.71877439685866626388071652128, 6.46362954514640221937282889832, 6.72104970897846474699859925754, 6.82890121268232047062070593990, 7.38887704653494610495339774320, 7.55993917983382946828184028512, 7.61329339867146147465158685342, 7.77534750555935756620399908332, 8.085896519930092250078765560673, 8.384196639544068837722160842646

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