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Properties

Label 8-1560e4-1.1-c3e4-0-0
Degree $8$
Conductor $5.922\times 10^{12}$
Sign $1$
Analytic cond. $7.17732\times 10^{7}$
Root an. cond. $9.59390$
Motivic weight $3$
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  + 12·3-s − 20·5-s + 15·7-s + 90·9-s − 7·11-s + 52·13-s − 240·15-s − 73·17-s + 68·19-s + 180·21-s − 49·23-s + 250·25-s + 540·27-s − 66·29-s + 230·31-s − 84·33-s − 300·35-s + 303·37-s + 624·39-s + 155·41-s + 336·43-s − 1.80e3·45-s + 178·47-s − 385·49-s − 876·51-s + 349·53-s + 140·55-s + ⋯
L(s)  = 1  + 2.30·3-s − 1.78·5-s + 0.809·7-s + 10/3·9-s − 0.191·11-s + 1.10·13-s − 4.13·15-s − 1.04·17-s + 0.821·19-s + 1.87·21-s − 0.444·23-s + 2·25-s + 3.84·27-s − 0.422·29-s + 1.33·31-s − 0.443·33-s − 1.44·35-s + 1.34·37-s + 2.56·39-s + 0.590·41-s + 1.19·43-s − 5.96·45-s + 0.552·47-s − 1.12·49-s − 2.40·51-s + 0.904·53-s + 0.343·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{12} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(7.17732\times 10^{7}\)
Root analytic conductor: \(9.59390\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{12} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(28.22225942\)
\(L(\frac12)\) \(\approx\) \(28.22225942\)
\(L(\frac{5}{2})\) 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)$
bad2 \( 1 \)
3$C_1$ \( ( 1 - p T )^{4} \)
5$C_1$ \( ( 1 + p T )^{4} \)
13$C_1$ \( ( 1 - p T )^{4} \)
good7$C_2 \wr S_4$ \( 1 - 15 T + 610 T^{2} - 1341 p T^{3} + 284250 T^{4} - 1341 p^{4} T^{5} + 610 p^{6} T^{6} - 15 p^{9} T^{7} + p^{12} T^{8} \)
11$C_2 \wr S_4$ \( 1 + 7 T + 788 T^{2} - 54209 T^{3} - 363866 T^{4} - 54209 p^{3} T^{5} + 788 p^{6} T^{6} + 7 p^{9} T^{7} + p^{12} T^{8} \)
17$C_2 \wr S_4$ \( 1 + 73 T + 17348 T^{2} + 1060927 T^{3} + 122423686 T^{4} + 1060927 p^{3} T^{5} + 17348 p^{6} T^{6} + 73 p^{9} T^{7} + p^{12} T^{8} \)
19$C_2 \wr S_4$ \( 1 - 68 T + 20416 T^{2} - 1128452 T^{3} + 189652366 T^{4} - 1128452 p^{3} T^{5} + 20416 p^{6} T^{6} - 68 p^{9} T^{7} + p^{12} T^{8} \)
23$C_2 \wr S_4$ \( 1 + 49 T + 45266 T^{2} + 1808197 T^{3} + 805661146 T^{4} + 1808197 p^{3} T^{5} + 45266 p^{6} T^{6} + 49 p^{9} T^{7} + p^{12} T^{8} \)
29$C_2 \wr S_4$ \( 1 + 66 T + 85328 T^{2} + 4186134 T^{3} + 2970993006 T^{4} + 4186134 p^{3} T^{5} + 85328 p^{6} T^{6} + 66 p^{9} T^{7} + p^{12} T^{8} \)
31$C_2 \wr S_4$ \( 1 - 230 T + 109828 T^{2} - 19812206 T^{3} + 4795719670 T^{4} - 19812206 p^{3} T^{5} + 109828 p^{6} T^{6} - 230 p^{9} T^{7} + p^{12} T^{8} \)
37$C_2 \wr S_4$ \( 1 - 303 T + 219766 T^{2} - 44921565 T^{3} + 17069345106 T^{4} - 44921565 p^{3} T^{5} + 219766 p^{6} T^{6} - 303 p^{9} T^{7} + p^{12} T^{8} \)
41$C_2 \wr S_4$ \( 1 - 155 T + 184766 T^{2} - 22013885 T^{3} + 16257572482 T^{4} - 22013885 p^{3} T^{5} + 184766 p^{6} T^{6} - 155 p^{9} T^{7} + p^{12} T^{8} \)
43$C_2 \wr S_4$ \( 1 - 336 T + 324268 T^{2} - 77850576 T^{3} + 904414242 p T^{4} - 77850576 p^{3} T^{5} + 324268 p^{6} T^{6} - 336 p^{9} T^{7} + p^{12} T^{8} \)
47$C_2 \wr S_4$ \( 1 - 178 T + 396284 T^{2} - 55260682 T^{3} + 60755876806 T^{4} - 55260682 p^{3} T^{5} + 396284 p^{6} T^{6} - 178 p^{9} T^{7} + p^{12} T^{8} \)
53$C_2 \wr S_4$ \( 1 - 349 T + 524210 T^{2} - 130449463 T^{3} + 111834728122 T^{4} - 130449463 p^{3} T^{5} + 524210 p^{6} T^{6} - 349 p^{9} T^{7} + p^{12} T^{8} \)
59$C_2 \wr S_4$ \( 1 + 360 T + 427004 T^{2} + 185567400 T^{3} + 121405569750 T^{4} + 185567400 p^{3} T^{5} + 427004 p^{6} T^{6} + 360 p^{9} T^{7} + p^{12} T^{8} \)
61$C_2 \wr S_4$ \( 1 - 223 T + 569458 T^{2} - 84435685 T^{3} + 164938925146 T^{4} - 84435685 p^{3} T^{5} + 569458 p^{6} T^{6} - 223 p^{9} T^{7} + p^{12} T^{8} \)
67$C_2 \wr S_4$ \( 1 - 588 T + 873772 T^{2} - 345473676 T^{3} + 349685675670 T^{4} - 345473676 p^{3} T^{5} + 873772 p^{6} T^{6} - 588 p^{9} T^{7} + p^{12} T^{8} \)
71$C_2 \wr S_4$ \( 1 + 83 T + 836780 T^{2} - 15362881 T^{3} + 347029137094 T^{4} - 15362881 p^{3} T^{5} + 836780 p^{6} T^{6} + 83 p^{9} T^{7} + p^{12} T^{8} \)
73$C_2 \wr S_4$ \( 1 - 754 T + 617296 T^{2} - 293439262 T^{3} + 126944659966 T^{4} - 293439262 p^{3} T^{5} + 617296 p^{6} T^{6} - 754 p^{9} T^{7} + p^{12} T^{8} \)
79$C_2 \wr S_4$ \( 1 - 11 p T + 984100 T^{2} - 964044833 T^{3} + 723922713334 T^{4} - 964044833 p^{3} T^{5} + 984100 p^{6} T^{6} - 11 p^{10} T^{7} + p^{12} T^{8} \)
83$C_2 \wr S_4$ \( 1 + 1044 T + 2321564 T^{2} + 1774723284 T^{3} + 2000090043510 T^{4} + 1774723284 p^{3} T^{5} + 2321564 p^{6} T^{6} + 1044 p^{9} T^{7} + p^{12} T^{8} \)
89$C_2 \wr S_4$ \( 1 - 1017 T + 1663262 T^{2} - 1415697039 T^{3} + 1357707670434 T^{4} - 1415697039 p^{3} T^{5} + 1663262 p^{6} T^{6} - 1017 p^{9} T^{7} + p^{12} T^{8} \)
97$C_2 \wr S_4$ \( 1 - 1367 T + 2301772 T^{2} - 1904633681 T^{3} + 2341938956470 T^{4} - 1904633681 p^{3} T^{5} + 2301772 p^{6} T^{6} - 1367 p^{9} T^{7} + p^{12} T^{8} \)
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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

−6.71299665495027736941443166157, −5.94491245981687299934001878803, −5.93139020431644711247735695280, −5.91191496243546066276797009003, −5.48155248539699662711163587888, −4.98658594598917044645159385573, −4.80288466265815051817659117140, −4.72540020001067192108062893023, −4.54384374264307171685058510842, −4.12783033498118367010984895369, −3.97879354278632255975080625296, −3.88910994678453111537513885684, −3.85364668142706608520937908920, −3.09771860579574954849915288498, −3.08922202847640162591328906835, −3.00370464873018555189127875768, −2.97737464927210586486172842067, −2.22031347270113825194863884767, −2.02604339203602168627322527497, −1.91403277182000521456956724027, −1.80769967334350033187099748666, −0.956677233613699777280516628020, −0.892899135186258587421945627216, −0.68711048288155108778909403881, −0.47435760369552245064640660973, 0.47435760369552245064640660973, 0.68711048288155108778909403881, 0.892899135186258587421945627216, 0.956677233613699777280516628020, 1.80769967334350033187099748666, 1.91403277182000521456956724027, 2.02604339203602168627322527497, 2.22031347270113825194863884767, 2.97737464927210586486172842067, 3.00370464873018555189127875768, 3.08922202847640162591328906835, 3.09771860579574954849915288498, 3.85364668142706608520937908920, 3.88910994678453111537513885684, 3.97879354278632255975080625296, 4.12783033498118367010984895369, 4.54384374264307171685058510842, 4.72540020001067192108062893023, 4.80288466265815051817659117140, 4.98658594598917044645159385573, 5.48155248539699662711163587888, 5.91191496243546066276797009003, 5.93139020431644711247735695280, 5.94491245981687299934001878803, 6.71299665495027736941443166157

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