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Properties

Label 8-1840e4-1.1-c3e4-0-0
Degree $8$
Conductor $1.146\times 10^{13}$
Sign $1$
Analytic cond. $1.38910\times 10^{8}$
Root an. cond. $10.4193$
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  + 4·3-s − 20·5-s + 7-s − 30·9-s + 39·11-s − 20·13-s − 80·15-s − 23·17-s − 53·19-s + 4·21-s + 92·23-s + 250·25-s − 223·27-s + 161·29-s − 388·31-s + 156·33-s − 20·35-s + 466·37-s − 80·39-s + 484·41-s − 894·43-s + 600·45-s + 265·47-s + 136·49-s − 92·51-s + 576·53-s − 780·55-s + ⋯
L(s)  = 1  + 0.769·3-s − 1.78·5-s + 0.0539·7-s − 1.11·9-s + 1.06·11-s − 0.426·13-s − 1.37·15-s − 0.328·17-s − 0.639·19-s + 0.0415·21-s + 0.834·23-s + 2·25-s − 1.58·27-s + 1.03·29-s − 2.24·31-s + 0.822·33-s − 0.0965·35-s + 2.07·37-s − 0.328·39-s + 1.84·41-s − 3.17·43-s + 1.98·45-s + 0.822·47-s + 0.396·49-s − 0.252·51-s + 1.49·53-s − 1.91·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 5^{4} \cdot 23^{4}\)
Sign: $1$
Analytic conductor: \(1.38910\times 10^{8}\)
Root analytic conductor: \(10.4193\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 5^{4} \cdot 23^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(7.417339399\)
\(L(\frac12)\) \(\approx\) \(7.417339399\)
\(L(\frac{5}{2})\) 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 \( 1 \)
5$C_1$ \( ( 1 + p T )^{4} \)
23$C_1$ \( ( 1 - p T )^{4} \)
good3$C_2 \wr S_4$ \( 1 - 4 T + 46 T^{2} - p^{4} T^{3} + 1190 T^{4} - p^{7} T^{5} + 46 p^{6} T^{6} - 4 p^{9} T^{7} + p^{12} T^{8} \)
7$C_2 \wr S_4$ \( 1 - T - 135 T^{2} - 521 p T^{3} + 125756 T^{4} - 521 p^{4} T^{5} - 135 p^{6} T^{6} - p^{9} T^{7} + p^{12} T^{8} \)
11$C_2 \wr S_4$ \( 1 - 39 T + 323 p T^{2} - 61307 T^{3} + 4937428 T^{4} - 61307 p^{3} T^{5} + 323 p^{7} T^{6} - 39 p^{9} T^{7} + p^{12} T^{8} \)
13$C_2 \wr S_4$ \( 1 + 20 T + 3840 T^{2} + 157301 T^{3} + 10278344 T^{4} + 157301 p^{3} T^{5} + 3840 p^{6} T^{6} + 20 p^{9} T^{7} + p^{12} T^{8} \)
17$C_2 \wr S_4$ \( 1 + 23 T + 5447 T^{2} - 251497 T^{3} + 6295268 T^{4} - 251497 p^{3} T^{5} + 5447 p^{6} T^{6} + 23 p^{9} T^{7} + p^{12} T^{8} \)
19$C_2 \wr S_4$ \( 1 + 53 T + 653 p T^{2} + 750501 T^{3} + 76029128 T^{4} + 750501 p^{3} T^{5} + 653 p^{7} T^{6} + 53 p^{9} T^{7} + p^{12} T^{8} \)
29$C_2 \wr S_4$ \( 1 - 161 T + 2424 p T^{2} - 9893175 T^{3} + 2240848710 T^{4} - 9893175 p^{3} T^{5} + 2424 p^{7} T^{6} - 161 p^{9} T^{7} + p^{12} T^{8} \)
31$C_2 \wr S_4$ \( 1 + 388 T + 132240 T^{2} + 29130015 T^{3} + 5707089166 T^{4} + 29130015 p^{3} T^{5} + 132240 p^{6} T^{6} + 388 p^{9} T^{7} + p^{12} T^{8} \)
37$C_2 \wr S_4$ \( 1 - 466 T + 240532 T^{2} - 71494750 T^{3} + 19227960950 T^{4} - 71494750 p^{3} T^{5} + 240532 p^{6} T^{6} - 466 p^{9} T^{7} + p^{12} T^{8} \)
41$C_2 \wr S_4$ \( 1 - 484 T + 239452 T^{2} - 82501249 T^{3} + 25012433496 T^{4} - 82501249 p^{3} T^{5} + 239452 p^{6} T^{6} - 484 p^{9} T^{7} + p^{12} T^{8} \)
43$C_2 \wr S_4$ \( 1 + 894 T + 484780 T^{2} + 190414350 T^{3} + 59052057270 T^{4} + 190414350 p^{3} T^{5} + 484780 p^{6} T^{6} + 894 p^{9} T^{7} + p^{12} T^{8} \)
47$C_2 \wr S_4$ \( 1 - 265 T + 184902 T^{2} - 39015297 T^{3} + 18141867922 T^{4} - 39015297 p^{3} T^{5} + 184902 p^{6} T^{6} - 265 p^{9} T^{7} + p^{12} T^{8} \)
53$C_2 \wr S_4$ \( 1 - 576 T + 488156 T^{2} - 177483648 T^{3} + 97785187798 T^{4} - 177483648 p^{3} T^{5} + 488156 p^{6} T^{6} - 576 p^{9} T^{7} + p^{12} T^{8} \)
59$C_2 \wr S_4$ \( 1 - 94 T + 578056 T^{2} - 37319902 T^{3} + 164753482782 T^{4} - 37319902 p^{3} T^{5} + 578056 p^{6} T^{6} - 94 p^{9} T^{7} + p^{12} T^{8} \)
61$C_2 \wr S_4$ \( 1 - 1153 T + 853867 T^{2} - 469738549 T^{3} + 241810092932 T^{4} - 469738549 p^{3} T^{5} + 853867 p^{6} T^{6} - 1153 p^{9} T^{7} + p^{12} T^{8} \)
67$C_2 \wr S_4$ \( 1 - 1472 T + 1805084 T^{2} - 1370871744 T^{3} + 894579577654 T^{4} - 1370871744 p^{3} T^{5} + 1805084 p^{6} T^{6} - 1472 p^{9} T^{7} + p^{12} T^{8} \)
71$C_2 \wr S_4$ \( 1 + 200 T + 364322 T^{2} + 121468265 T^{3} + 278790401066 T^{4} + 121468265 p^{3} T^{5} + 364322 p^{6} T^{6} + 200 p^{9} T^{7} + p^{12} T^{8} \)
73$C_2 \wr S_4$ \( 1 - 1147 T + 1912188 T^{2} - 1347657401 T^{3} + 1180324718270 T^{4} - 1347657401 p^{3} T^{5} + 1912188 p^{6} T^{6} - 1147 p^{9} T^{7} + p^{12} T^{8} \)
79$C_2 \wr S_4$ \( 1 - 908 T + 1048492 T^{2} - 1070426012 T^{3} + 693505308902 T^{4} - 1070426012 p^{3} T^{5} + 1048492 p^{6} T^{6} - 908 p^{9} T^{7} + p^{12} T^{8} \)
83$C_2 \wr S_4$ \( 1 - 1048 T + 1927704 T^{2} - 1122552888 T^{3} + 1371917773086 T^{4} - 1122552888 p^{3} T^{5} + 1927704 p^{6} T^{6} - 1048 p^{9} T^{7} + p^{12} T^{8} \)
89$C_2 \wr S_4$ \( 1 + 1784 T + 2327792 T^{2} + 1523808488 T^{3} + 1318662054974 T^{4} + 1523808488 p^{3} T^{5} + 2327792 p^{6} T^{6} + 1784 p^{9} T^{7} + p^{12} T^{8} \)
97$C_2 \wr S_4$ \( 1 + 2047 T + 2933893 T^{2} + 2857813609 T^{3} + 3031159147844 T^{4} + 2857813609 p^{3} T^{5} + 2933893 p^{6} T^{6} + 2047 p^{9} T^{7} + p^{12} T^{8} \)
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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

−6.43281136248278194634687419635, −5.96009959231081196906180317465, −5.77886825034172779044131300122, −5.47970573799921166060723577870, −5.34597739352488599190168211991, −5.04413336384957841475612249174, −5.00467325095024964846175845635, −4.65147430745636089967657581300, −4.38809472123368319219420408334, −3.93769504048740597667779278193, −3.88967382860568436464018399220, −3.88431606693556110938599595462, −3.77826805176956558262818533438, −3.32705388150961290378074526129, −2.92623676747851056317283656707, −2.87092113838626018366321682818, −2.72470672805207909512726559122, −2.23482705976915827804534882134, −2.10758984405516018665530624379, −1.89358324254576197564126538384, −1.41878223218199821610406992081, −0.987318092020455112088475131885, −0.55595890494609946138227564017, −0.49462493273344896557532024330, −0.48929196218766658848787034185, 0.48929196218766658848787034185, 0.49462493273344896557532024330, 0.55595890494609946138227564017, 0.987318092020455112088475131885, 1.41878223218199821610406992081, 1.89358324254576197564126538384, 2.10758984405516018665530624379, 2.23482705976915827804534882134, 2.72470672805207909512726559122, 2.87092113838626018366321682818, 2.92623676747851056317283656707, 3.32705388150961290378074526129, 3.77826805176956558262818533438, 3.88431606693556110938599595462, 3.88967382860568436464018399220, 3.93769504048740597667779278193, 4.38809472123368319219420408334, 4.65147430745636089967657581300, 5.00467325095024964846175845635, 5.04413336384957841475612249174, 5.34597739352488599190168211991, 5.47970573799921166060723577870, 5.77886825034172779044131300122, 5.96009959231081196906180317465, 6.43281136248278194634687419635

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