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Properties

Label 8-570e4-1.1-c3e4-0-2
Degree $8$
Conductor $105560010000$
Sign $1$
Analytic cond. $1.27927\times 10^{6}$
Root an. cond. $5.79923$
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  + 8·2-s + 12·3-s + 40·4-s + 20·5-s + 96·6-s + 36·7-s + 160·8-s + 90·9-s + 160·10-s + 54·11-s + 480·12-s + 46·13-s + 288·14-s + 240·15-s + 560·16-s + 14·17-s + 720·18-s + 76·19-s + 800·20-s + 432·21-s + 432·22-s + 104·23-s + 1.92e3·24-s + 250·25-s + 368·26-s + 540·27-s + 1.44e3·28-s + ⋯
L(s)  = 1  + 2.82·2-s + 2.30·3-s + 5·4-s + 1.78·5-s + 6.53·6-s + 1.94·7-s + 7.07·8-s + 10/3·9-s + 5.05·10-s + 1.48·11-s + 11.5·12-s + 0.981·13-s + 5.49·14-s + 4.13·15-s + 35/4·16-s + 0.199·17-s + 9.42·18-s + 0.917·19-s + 8.94·20-s + 4.48·21-s + 4.18·22-s + 0.942·23-s + 16.3·24-s + 2·25-s + 2.77·26-s + 3.84·27-s + 9.71·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 19^{4}\)
Sign: $1$
Analytic conductor: \(1.27927\times 10^{6}\)
Root analytic conductor: \(5.79923\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 19^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(490.0721825\)
\(L(\frac12)\) \(\approx\) \(490.0721825\)
\(L(\frac{5}{2})\) 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$C_1$ \( ( 1 - p T )^{4} \)
3$C_1$ \( ( 1 - p T )^{4} \)
5$C_1$ \( ( 1 - p T )^{4} \)
19$C_1$ \( ( 1 - p T )^{4} \)
good7$C_2 \wr S_4$ \( 1 - 36 T + 1094 T^{2} - 27528 T^{3} + 529218 T^{4} - 27528 p^{3} T^{5} + 1094 p^{6} T^{6} - 36 p^{9} T^{7} + p^{12} T^{8} \)
11$C_2 \wr S_4$ \( 1 - 54 T + 4754 T^{2} - 178974 T^{3} + 9146586 T^{4} - 178974 p^{3} T^{5} + 4754 p^{6} T^{6} - 54 p^{9} T^{7} + p^{12} T^{8} \)
13$C_2 \wr S_4$ \( 1 - 46 T + 4958 T^{2} - 4074 p T^{3} + 8724818 T^{4} - 4074 p^{4} T^{5} + 4958 p^{6} T^{6} - 46 p^{9} T^{7} + p^{12} T^{8} \)
17$C_2 \wr S_4$ \( 1 - 14 T + 5436 T^{2} - 22890 T^{3} + 46774966 T^{4} - 22890 p^{3} T^{5} + 5436 p^{6} T^{6} - 14 p^{9} T^{7} + p^{12} T^{8} \)
23$C_2 \wr S_4$ \( 1 - 104 T + 41472 T^{2} - 161448 p T^{3} + 31052690 p T^{4} - 161448 p^{4} T^{5} + 41472 p^{6} T^{6} - 104 p^{9} T^{7} + p^{12} T^{8} \)
29$C_2 \wr S_4$ \( 1 - 14 T + 24246 T^{2} + 6190902 T^{3} - 188087054 T^{4} + 6190902 p^{3} T^{5} + 24246 p^{6} T^{6} - 14 p^{9} T^{7} + p^{12} T^{8} \)
31$C_2 \wr S_4$ \( 1 - 30 T + 67256 T^{2} - 1138830 T^{3} + 2248409166 T^{4} - 1138830 p^{3} T^{5} + 67256 p^{6} T^{6} - 30 p^{9} T^{7} + p^{12} T^{8} \)
37$C_2 \wr S_4$ \( 1 - 30 T + 155126 T^{2} - 2078322 T^{3} + 10598777730 T^{4} - 2078322 p^{3} T^{5} + 155126 p^{6} T^{6} - 30 p^{9} T^{7} + p^{12} T^{8} \)
41$C_2 \wr S_4$ \( 1 + 36 T + 40838 T^{2} + 5900904 T^{3} + 8187891162 T^{4} + 5900904 p^{3} T^{5} + 40838 p^{6} T^{6} + 36 p^{9} T^{7} + p^{12} T^{8} \)
43$C_2 \wr S_4$ \( 1 - 102 T + 155450 T^{2} - 11230854 T^{3} + 17032833738 T^{4} - 11230854 p^{3} T^{5} + 155450 p^{6} T^{6} - 102 p^{9} T^{7} + p^{12} T^{8} \)
47$C_2 \wr S_4$ \( 1 - 408 T + 37664 T^{2} + 34682328 T^{3} - 12654742914 T^{4} + 34682328 p^{3} T^{5} + 37664 p^{6} T^{6} - 408 p^{9} T^{7} + p^{12} T^{8} \)
53$C_2 \wr S_4$ \( 1 + 176 T + 221568 T^{2} + 86956272 T^{3} + 38996047918 T^{4} + 86956272 p^{3} T^{5} + 221568 p^{6} T^{6} + 176 p^{9} T^{7} + p^{12} T^{8} \)
59$C_2 \wr S_4$ \( 1 - 66 T + 613028 T^{2} + 14457150 T^{3} + 163724073222 T^{4} + 14457150 p^{3} T^{5} + 613028 p^{6} T^{6} - 66 p^{9} T^{7} + p^{12} T^{8} \)
61$C_2 \wr S_4$ \( 1 - 60 T + 449720 T^{2} + 87467724 T^{3} + 95146306590 T^{4} + 87467724 p^{3} T^{5} + 449720 p^{6} T^{6} - 60 p^{9} T^{7} + p^{12} T^{8} \)
67$C_2 \wr S_4$ \( 1 + 152 T + 397964 T^{2} - 214730856 T^{3} + 36703872470 T^{4} - 214730856 p^{3} T^{5} + 397964 p^{6} T^{6} + 152 p^{9} T^{7} + p^{12} T^{8} \)
71$C_2 \wr S_4$ \( 1 - 172 T + 1345524 T^{2} - 175034076 T^{3} + 707293032886 T^{4} - 175034076 p^{3} T^{5} + 1345524 p^{6} T^{6} - 172 p^{9} T^{7} + p^{12} T^{8} \)
73$C_2 \wr S_4$ \( 1 - 284 T + 790852 T^{2} - 69814724 T^{3} + 365062411990 T^{4} - 69814724 p^{3} T^{5} + 790852 p^{6} T^{6} - 284 p^{9} T^{7} + p^{12} T^{8} \)
79$C_2 \wr S_4$ \( 1 - 554 T + 1701580 T^{2} - 722604578 T^{3} + 1217920492198 T^{4} - 722604578 p^{3} T^{5} + 1701580 p^{6} T^{6} - 554 p^{9} T^{7} + p^{12} T^{8} \)
83$C_2 \wr S_4$ \( 1 + 394 T + 2160096 T^{2} + 647054034 T^{3} + 1822415277646 T^{4} + 647054034 p^{3} T^{5} + 2160096 p^{6} T^{6} + 394 p^{9} T^{7} + p^{12} T^{8} \)
89$C_2 \wr S_4$ \( 1 + 60 T - 544906 T^{2} - 177585624 T^{3} + 680167453770 T^{4} - 177585624 p^{3} T^{5} - 544906 p^{6} T^{6} + 60 p^{9} T^{7} + p^{12} T^{8} \)
97$C_2 \wr S_4$ \( 1 + 922 T + 8830 p T^{2} + 260251822 T^{3} + 892373198482 T^{4} + 260251822 p^{3} T^{5} + 8830 p^{7} T^{6} + 922 p^{9} T^{7} + p^{12} T^{8} \)
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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

−7.36729053130512523589411828453, −6.77254714027759588058804954052, −6.67763880506188632599863711077, −6.61374144484389916926021909704, −6.51489857655699754147568154162, −5.86394705474324688459683045794, −5.73215814991406824817185981115, −5.49915073002812605586092607198, −5.46537821916481503205601926367, −4.76911385177287180208085360296, −4.68497863657921401920452919508, −4.66091294546231502271803440339, −4.49211862605399354678890965560, −3.74503850301461798801606447490, −3.59564441711580344733389554456, −3.58420468777528280784314531030, −3.49120062546264648013116712939, −2.60475633422256225250849077622, −2.60344413882605254454592564940, −2.43132578258102578545455861240, −2.23329894503248346775294322008, −1.49383637740674338184212860477, −1.35370683666873361501335739651, −1.24556100229978945553509734081, −1.22168469564904644619736435155, 1.22168469564904644619736435155, 1.24556100229978945553509734081, 1.35370683666873361501335739651, 1.49383637740674338184212860477, 2.23329894503248346775294322008, 2.43132578258102578545455861240, 2.60344413882605254454592564940, 2.60475633422256225250849077622, 3.49120062546264648013116712939, 3.58420468777528280784314531030, 3.59564441711580344733389554456, 3.74503850301461798801606447490, 4.49211862605399354678890965560, 4.66091294546231502271803440339, 4.68497863657921401920452919508, 4.76911385177287180208085360296, 5.46537821916481503205601926367, 5.49915073002812605586092607198, 5.73215814991406824817185981115, 5.86394705474324688459683045794, 6.51489857655699754147568154162, 6.61374144484389916926021909704, 6.67763880506188632599863711077, 6.77254714027759588058804954052, 7.36729053130512523589411828453

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