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Properties

Label 8-1280e4-1.1-c3e4-0-1
Degree $8$
Conductor $2.684\times 10^{12}$
Sign $1$
Analytic cond. $3.25315\times 10^{7}$
Root an. cond. $8.69036$
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  − 20·5-s + 32·7-s − 28·9-s − 16·11-s − 80·13-s − 48·17-s − 208·19-s + 208·23-s + 250·25-s − 48·27-s − 160·29-s + 32·31-s − 640·35-s + 216·37-s − 104·41-s − 304·43-s + 560·45-s − 608·47-s − 364·49-s + 720·53-s + 320·55-s + 48·59-s − 1.52e3·61-s − 896·63-s + 1.60e3·65-s + 16·67-s + 672·71-s + ⋯
L(s)  = 1  − 1.78·5-s + 1.72·7-s − 1.03·9-s − 0.438·11-s − 1.70·13-s − 0.684·17-s − 2.51·19-s + 1.88·23-s + 2·25-s − 0.342·27-s − 1.02·29-s + 0.185·31-s − 3.09·35-s + 0.959·37-s − 0.396·41-s − 1.07·43-s + 1.85·45-s − 1.88·47-s − 1.06·49-s + 1.86·53-s + 0.784·55-s + 0.105·59-s − 3.20·61-s − 1.79·63-s + 3.05·65-s + 0.0291·67-s + 1.12·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{32} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(3.25315\times 10^{7}\)
Root analytic conductor: \(8.69036\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{32} \cdot 5^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(0.1800672050\)
\(L(\frac12)\) \(\approx\) \(0.1800672050\)
\(L(\frac{5}{2})\) 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$C_1$ \( ( 1 + p T )^{4} \)
good3$C_2 \wr C_2\wr C_2$ \( 1 + 28 T^{2} + 16 p T^{3} + 1162 T^{4} + 16 p^{4} T^{5} + 28 p^{6} T^{6} + p^{12} T^{8} \)
7$C_2 \wr C_2\wr C_2$ \( 1 - 32 T + 1388 T^{2} - 28752 T^{3} + 706778 T^{4} - 28752 p^{3} T^{5} + 1388 p^{6} T^{6} - 32 p^{9} T^{7} + p^{12} T^{8} \)
11$C_2 \wr C_2\wr C_2$ \( 1 + 16 T + 1236 T^{2} - 36144 T^{3} + 348982 T^{4} - 36144 p^{3} T^{5} + 1236 p^{6} T^{6} + 16 p^{9} T^{7} + p^{12} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 + 80 T + 6332 T^{2} + 2160 p^{2} T^{3} + 19958774 T^{4} + 2160 p^{5} T^{5} + 6332 p^{6} T^{6} + 80 p^{9} T^{7} + p^{12} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 + 48 T + 6380 T^{2} + 1872 p^{2} T^{3} + 14684454 T^{4} + 1872 p^{5} T^{5} + 6380 p^{6} T^{6} + 48 p^{9} T^{7} + p^{12} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 + 208 T + 29828 T^{2} + 3019920 T^{3} + 270314486 T^{4} + 3019920 p^{3} T^{5} + 29828 p^{6} T^{6} + 208 p^{9} T^{7} + p^{12} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 - 208 T + 53916 T^{2} - 6542688 T^{3} + 965949466 T^{4} - 6542688 p^{3} T^{5} + 53916 p^{6} T^{6} - 208 p^{9} T^{7} + p^{12} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 + 160 T + 71508 T^{2} + 6914400 T^{3} + 2121423382 T^{4} + 6914400 p^{3} T^{5} + 71508 p^{6} T^{6} + 160 p^{9} T^{7} + p^{12} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 - 32 T + 62780 T^{2} - 1351200 T^{3} + 2540467526 T^{4} - 1351200 p^{3} T^{5} + 62780 p^{6} T^{6} - 32 p^{9} T^{7} + p^{12} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 - 216 T + 77644 T^{2} + 7885368 T^{3} - 322909002 T^{4} + 7885368 p^{3} T^{5} + 77644 p^{6} T^{6} - 216 p^{9} T^{7} + p^{12} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 + 104 T + 2628 p T^{2} - 583368 T^{3} + 4695256678 T^{4} - 583368 p^{3} T^{5} + 2628 p^{7} T^{6} + 104 p^{9} T^{7} + p^{12} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 + 304 T + 257564 T^{2} + 60851232 T^{3} + 28133666666 T^{4} + 60851232 p^{3} T^{5} + 257564 p^{6} T^{6} + 304 p^{9} T^{7} + p^{12} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 + 608 T + 7044 p T^{2} + 136705776 T^{3} + 54510789754 T^{4} + 136705776 p^{3} T^{5} + 7044 p^{7} T^{6} + 608 p^{9} T^{7} + p^{12} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 - 720 T + 604316 T^{2} - 279540720 T^{3} + 136365680406 T^{4} - 279540720 p^{3} T^{5} + 604316 p^{6} T^{6} - 720 p^{9} T^{7} + p^{12} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 - 48 T + 644708 T^{2} - 32690160 T^{3} + 184670949846 T^{4} - 32690160 p^{3} T^{5} + 644708 p^{6} T^{6} - 48 p^{9} T^{7} + p^{12} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 + 1528 T + 947660 T^{2} + 271898088 T^{3} + 61093374422 T^{4} + 271898088 p^{3} T^{5} + 947660 p^{6} T^{6} + 1528 p^{9} T^{7} + p^{12} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 - 16 T + 964988 T^{2} - 17138496 T^{3} + 404649745418 T^{4} - 17138496 p^{3} T^{5} + 964988 p^{6} T^{6} - 16 p^{9} T^{7} + p^{12} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 - 672 T + 487964 T^{2} + 79682400 T^{3} - 9249307098 T^{4} + 79682400 p^{3} T^{5} + 487964 p^{6} T^{6} - 672 p^{9} T^{7} + p^{12} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 + 288 T + 1217708 T^{2} + 258840288 T^{3} + 650251220166 T^{4} + 258840288 p^{3} T^{5} + 1217708 p^{6} T^{6} + 288 p^{9} T^{7} + p^{12} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 - 1856 T + 2445212 T^{2} - 2258762304 T^{3} + 1748697322502 T^{4} - 2258762304 p^{3} T^{5} + 2445212 p^{6} T^{6} - 1856 p^{9} T^{7} + p^{12} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 + 2672 T + 4145772 T^{2} + 4653569952 T^{3} + 4047232375690 T^{4} + 4653569952 p^{3} T^{5} + 4145772 p^{6} T^{6} + 2672 p^{9} T^{7} + p^{12} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 + 648 T + 1609820 T^{2} + 281611512 T^{3} + 1101610526310 T^{4} + 281611512 p^{3} T^{5} + 1609820 p^{6} T^{6} + 648 p^{9} T^{7} + p^{12} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 + 448 T + 2154764 T^{2} + 628047936 T^{3} + 2536331669414 T^{4} + 628047936 p^{3} T^{5} + 2154764 p^{6} T^{6} + 448 p^{9} T^{7} + p^{12} T^{8} \)
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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

−6.76608087331614421676428483327, −6.20233457532758732262097741353, −6.08115080334122022372908690396, −5.86111652099944499325310432984, −5.61626778427616400963138322144, −5.24288771154812497103674453182, −5.05145817179684619908723429066, −4.91072368173741107770443387892, −4.56901619444248913250991533613, −4.55538391964869156978301750666, −4.38641553594516718494145883605, −4.16211006128153809038865400314, −3.90522010186078371697941565649, −3.21255255132874928093882305855, −3.15007765057272523828703104336, −3.06157045660935114542805511633, −3.00717034601482116750057184034, −2.25808150202252188834750539314, −2.02759193749771149926379041846, −1.96042540799142538762027378868, −1.75563345171666224651043565777, −1.15231380989184524734210926177, −0.75217261635611731874148911984, −0.34713594388026964248845513699, −0.093908650457124717685358543853, 0.093908650457124717685358543853, 0.34713594388026964248845513699, 0.75217261635611731874148911984, 1.15231380989184524734210926177, 1.75563345171666224651043565777, 1.96042540799142538762027378868, 2.02759193749771149926379041846, 2.25808150202252188834750539314, 3.00717034601482116750057184034, 3.06157045660935114542805511633, 3.15007765057272523828703104336, 3.21255255132874928093882305855, 3.90522010186078371697941565649, 4.16211006128153809038865400314, 4.38641553594516718494145883605, 4.55538391964869156978301750666, 4.56901619444248913250991533613, 4.91072368173741107770443387892, 5.05145817179684619908723429066, 5.24288771154812497103674453182, 5.61626778427616400963138322144, 5.86111652099944499325310432984, 6.08115080334122022372908690396, 6.20233457532758732262097741353, 6.76608087331614421676428483327

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