Properties

Label 8-720e4-1.1-c2e4-0-4
Degree $8$
Conductor $268738560000$
Sign $1$
Analytic cond. $148139.$
Root an. cond. $4.42928$
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  + 16·7-s + 8·13-s − 32·19-s − 10·25-s − 8·31-s − 136·37-s − 80·43-s + 144·49-s − 40·61-s + 304·67-s + 152·73-s − 200·79-s + 128·91-s − 424·97-s + 112·103-s + 104·109-s + 88·121-s + 127-s + 131-s − 512·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  + 16/7·7-s + 8/13·13-s − 1.68·19-s − 2/5·25-s − 0.258·31-s − 3.67·37-s − 1.86·43-s + 2.93·49-s − 0.655·61-s + 4.53·67-s + 2.08·73-s − 2.53·79-s + 1.40·91-s − 4.37·97-s + 1.08·103-s + 0.954·109-s + 8/11·121-s + 0.00787·127-s + 0.00763·131-s − 3.84·133-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 3^{8} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(148139.\)
Root analytic conductor: \(4.42928\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 3^{8} \cdot 5^{4} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.6848023655\)
\(L(\frac12)\) \(\approx\) \(0.6848023655\)
\(L(2)\) 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 \( 1 \)
5$C_2$ \( ( 1 + p T^{2} )^{2} \)
good7$D_{4}$ \( ( 1 - 8 T + 24 T^{2} - 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
11$D_4\times C_2$ \( 1 - 8 p T^{2} + 18258 T^{4} - 8 p^{5} T^{6} + p^{8} T^{8} \)
13$D_{4}$ \( ( 1 - 4 T + 252 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 - 290 T^{2} + p^{4} T^{4} )^{2} \)
19$D_{4}$ \( ( 1 + 16 T + 426 T^{2} + 16 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 - 1612 T^{2} + 1157478 T^{4} - 1612 p^{4} T^{6} + p^{8} T^{8} \)
29$D_4\times C_2$ \( 1 - 700 T^{2} + 707622 T^{4} - 700 p^{4} T^{6} + p^{8} T^{8} \)
31$D_{4}$ \( ( 1 + 4 T + 1566 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
37$D_{4}$ \( ( 1 + 68 T + 3084 T^{2} + 68 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
41$D_4\times C_2$ \( 1 - 640 T^{2} + 3887682 T^{4} - 640 p^{4} T^{6} + p^{8} T^{8} \)
43$D_{4}$ \( ( 1 + 40 T + 3738 T^{2} + 40 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
47$D_4\times C_2$ \( 1 - 340 T^{2} - 372378 T^{4} - 340 p^{4} T^{6} + p^{8} T^{8} \)
53$D_4\times C_2$ \( 1 - 1876 T^{2} - 4075194 T^{4} - 1876 p^{4} T^{6} + p^{8} T^{8} \)
59$D_4\times C_2$ \( 1 - 280 T^{2} - 9454638 T^{4} - 280 p^{4} T^{6} + p^{8} T^{8} \)
61$D_{4}$ \( ( 1 + 20 T + 4302 T^{2} + 20 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
67$C_2$ \( ( 1 - 76 T + p^{2} T^{2} )^{4} \)
71$D_4\times C_2$ \( 1 - 13540 T^{2} + 89191302 T^{4} - 13540 p^{4} T^{6} + p^{8} T^{8} \)
73$D_{4}$ \( ( 1 - 76 T + 8862 T^{2} - 76 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
79$D_{4}$ \( ( 1 + 100 T + 11742 T^{2} + 100 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - 21652 T^{2} + 211289478 T^{4} - 21652 p^{4} T^{6} + p^{8} T^{8} \)
89$D_4\times C_2$ \( 1 - 14368 T^{2} + 154232898 T^{4} - 14368 p^{4} T^{6} + p^{8} T^{8} \)
97$D_{4}$ \( ( 1 + 212 T + 29694 T^{2} + 212 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
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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

−7.04476849637066490497916188646, −6.98963220065363967222606741816, −6.95934374592816069682582675446, −6.69714931510391134106620605117, −6.31133656697192482564604904614, −6.22899436201162062239096346755, −5.61352667230731727304522876065, −5.54994244135412189729308303437, −5.38707631592769120843601027603, −5.09093116128727799445949974509, −4.87522221266028126344407030500, −4.78686215429908225783312099447, −4.13436518058791019141432267009, −4.13300521526022685067282386340, −4.01130569334386354094803118491, −3.47906620245605414913583556481, −3.34922981298840503628692610327, −2.99358628069943407213453880940, −2.30742842519357916870965460479, −2.21001527669825214598935552310, −1.95816018837077568326704633334, −1.57156501231235086105516507672, −1.39091026862867481615309934523, −0.877783202744820745209582979923, −0.12597981920862756534044671369, 0.12597981920862756534044671369, 0.877783202744820745209582979923, 1.39091026862867481615309934523, 1.57156501231235086105516507672, 1.95816018837077568326704633334, 2.21001527669825214598935552310, 2.30742842519357916870965460479, 2.99358628069943407213453880940, 3.34922981298840503628692610327, 3.47906620245605414913583556481, 4.01130569334386354094803118491, 4.13300521526022685067282386340, 4.13436518058791019141432267009, 4.78686215429908225783312099447, 4.87522221266028126344407030500, 5.09093116128727799445949974509, 5.38707631592769120843601027603, 5.54994244135412189729308303437, 5.61352667230731727304522876065, 6.22899436201162062239096346755, 6.31133656697192482564604904614, 6.69714931510391134106620605117, 6.95934374592816069682582675446, 6.98963220065363967222606741816, 7.04476849637066490497916188646

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