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Properties

Label 8-75e4-1.1-c8e4-0-0
Degree $8$
Conductor $31640625$
Sign $1$
Analytic cond. $871440.$
Root an. cond. $5.52751$
Motivic weight $8$
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  − 936·11-s − 3.21e4·16-s + 3.34e6·31-s + 1.12e7·41-s + 2.06e6·61-s − 8.33e7·71-s − 4.78e6·81-s − 3.45e8·101-s − 8.56e8·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 3.00e7·176-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  − 0.0639·11-s − 0.490·16-s + 3.62·31-s + 3.99·41-s + 0.149·61-s − 3.27·71-s − 1/9·81-s − 3.31·101-s − 3.99·121-s + 0.0313·176-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s+4)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(31640625\)    =    \(3^{4} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(871440.\)
Root analytic conductor: \(5.52751\)
Motivic weight: \(8\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 31640625,\ (\ :4, 4, 4, 4),\ 1)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.02960346960\)
\(L(\frac12)\) \(\approx\) \(0.02960346960\)
\(L(5)\) 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$C_2^2$ \( 1 + p^{14} T^{4} \)
5 \( 1 \)
good2$C_2^3$ \( 1 + 2009 p^{4} T^{4} + p^{32} T^{8} \)
7$C_2^3$ \( 1 - 17405278273 p^{4} T^{4} + p^{32} T^{8} \)
11$C_2$ \( ( 1 + 234 T + p^{8} T^{2} )^{4} \)
13$C_2^3$ \( 1 + 483037557660079919 T^{4} + p^{32} T^{8} \)
17$C_2^3$ \( 1 - 93732267649515308062 T^{4} + p^{32} T^{8} \)
19$C_2^2$ \( ( 1 - 954779833 T^{2} + p^{16} T^{4} )^{2} \)
23$C_2^3$ \( 1 - \)\(12\!\cdots\!78\)\( T^{4} + p^{32} T^{8} \)
29$C_2^2$ \( ( 1 - 942809275646 T^{2} + p^{16} T^{4} )^{2} \)
31$C_2$ \( ( 1 - 836725 T + p^{8} T^{2} )^{4} \)
37$C_2^3$ \( 1 + \)\(14\!\cdots\!82\)\( T^{4} + p^{32} T^{8} \)
41$C_2$ \( ( 1 - 2822220 T + p^{8} T^{2} )^{4} \)
43$C_2^3$ \( 1 - \)\(21\!\cdots\!77\)\( T^{4} + p^{32} T^{8} \)
47$C_2^3$ \( 1 - \)\(10\!\cdots\!42\)\( T^{4} + p^{32} T^{8} \)
53$C_2^3$ \( 1 + \)\(70\!\cdots\!42\)\( T^{4} + p^{32} T^{8} \)
59$C_2^2$ \( ( 1 - 130180153230878 T^{2} + p^{16} T^{4} )^{2} \)
61$C_2$ \( ( 1 - 517403 T + p^{8} T^{2} )^{4} \)
67$C_2^3$ \( 1 + \)\(31\!\cdots\!63\)\( T^{4} + p^{32} T^{8} \)
71$C_2$ \( ( 1 + 20828628 T + p^{8} T^{2} )^{4} \)
73$C_2^3$ \( 1 - \)\(12\!\cdots\!78\)\( T^{4} + p^{32} T^{8} \)
79$C_2^2$ \( ( 1 - 1254396182128222 T^{2} + p^{16} T^{4} )^{2} \)
83$C_2^3$ \( 1 + \)\(90\!\cdots\!34\)\( T^{4} + p^{32} T^{8} \)
89$C_2^2$ \( ( 1 + 1182519767595022 T^{2} + p^{16} T^{4} )^{2} \)
97$C_2^3$ \( 1 - \)\(91\!\cdots\!17\)\( T^{4} + p^{32} T^{8} \)
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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

−9.060642568539497555725260417764, −8.963150585127433837992245385777, −8.192067298856133163243548694877, −8.155687464069972483373543787921, −7.964454907992431261094182434449, −7.57360157801770434175578266151, −7.11244781900148083908278255789, −6.85865016063267525644476247027, −6.55981494139943254658019173615, −6.04171746290322347558079540539, −6.00163013362197441015295724721, −5.59472730288084429719634403542, −5.13125732194581415319086958823, −4.58886396829566901857450112384, −4.47212726158562086098295036631, −4.13445994986186759614583914155, −3.83246254523478569494325519916, −2.97034052074813939067322676274, −2.89585191350432314119044518954, −2.45555746967514716435869207084, −2.26725984284773535107296067855, −1.29830175434746065403103507239, −1.17152137032661375207576927699, −0.866601878202928419442388001279, −0.02431373328381196371886089797, 0.02431373328381196371886089797, 0.866601878202928419442388001279, 1.17152137032661375207576927699, 1.29830175434746065403103507239, 2.26725984284773535107296067855, 2.45555746967514716435869207084, 2.89585191350432314119044518954, 2.97034052074813939067322676274, 3.83246254523478569494325519916, 4.13445994986186759614583914155, 4.47212726158562086098295036631, 4.58886396829566901857450112384, 5.13125732194581415319086958823, 5.59472730288084429719634403542, 6.00163013362197441015295724721, 6.04171746290322347558079540539, 6.55981494139943254658019173615, 6.85865016063267525644476247027, 7.11244781900148083908278255789, 7.57360157801770434175578266151, 7.964454907992431261094182434449, 8.155687464069972483373543787921, 8.192067298856133163243548694877, 8.963150585127433837992245385777, 9.060642568539497555725260417764

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