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Properties

Label 8-162e4-1.1-c3e4-0-1
Degree $8$
Conductor $688747536$
Sign $1$
Analytic cond. $8346.88$
Root an. cond. $3.09165$
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  + 4·2-s + 4·4-s + 12·5-s − 16·7-s − 16·8-s + 48·10-s + 36·11-s − 10·13-s − 64·14-s − 64·16-s − 240·17-s − 16·19-s + 48·20-s + 144·22-s + 180·23-s + 259·25-s − 40·26-s − 64·28-s + 324·29-s + 248·31-s − 64·32-s − 960·34-s − 192·35-s − 868·37-s − 64·38-s − 192·40-s + 192·41-s + ⋯
L(s)  = 1  + 1.41·2-s + 1/2·4-s + 1.07·5-s − 0.863·7-s − 0.707·8-s + 1.51·10-s + 0.986·11-s − 0.213·13-s − 1.22·14-s − 16-s − 3.42·17-s − 0.193·19-s + 0.536·20-s + 1.39·22-s + 1.63·23-s + 2.07·25-s − 0.301·26-s − 0.431·28-s + 2.07·29-s + 1.43·31-s − 0.353·32-s − 4.84·34-s − 0.927·35-s − 3.85·37-s − 0.273·38-s − 0.758·40-s + 0.731·41-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{16}\)
Sign: $1$
Analytic conductor: \(8346.88\)
Root analytic conductor: \(3.09165\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{16} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(8.455956000\)
\(L(\frac12)\) \(\approx\) \(8.455956000\)
\(L(\frac{5}{2})\) 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$C_2$ \( ( 1 - p T + p^{2} T^{2} )^{2} \)
3 \( 1 \)
good5$D_4\times C_2$ \( 1 - 12 T - 23 p T^{2} - 108 T^{3} + 33456 T^{4} - 108 p^{3} T^{5} - 23 p^{7} T^{6} - 12 p^{9} T^{7} + p^{12} T^{8} \)
7$D_4\times C_2$ \( 1 + 16 T - 386 T^{2} - 704 T^{3} + 206707 T^{4} - 704 p^{3} T^{5} - 386 p^{6} T^{6} + 16 p^{9} T^{7} + p^{12} T^{8} \)
11$D_4\times C_2$ \( 1 - 36 T + 38 T^{2} + 50544 T^{3} - 1913973 T^{4} + 50544 p^{3} T^{5} + 38 p^{6} T^{6} - 36 p^{9} T^{7} + p^{12} T^{8} \)
13$D_4\times C_2$ \( 1 + 10 T + 973 T^{2} - 52670 T^{3} - 4284380 T^{4} - 52670 p^{3} T^{5} + 973 p^{6} T^{6} + 10 p^{9} T^{7} + p^{12} T^{8} \)
17$D_{4}$ \( ( 1 + 120 T + 10159 T^{2} + 120 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
19$D_{4}$ \( ( 1 + 8 T + 666 T^{2} + 8 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 - 180 T + 398 T^{2} - 1380240 T^{3} + 481881315 T^{4} - 1380240 p^{3} T^{5} + 398 p^{6} T^{6} - 180 p^{9} T^{7} + p^{12} T^{8} \)
29$D_4\times C_2$ \( 1 - 324 T + 31277 T^{2} - 8074404 T^{3} + 2276459616 T^{4} - 8074404 p^{3} T^{5} + 31277 p^{6} T^{6} - 324 p^{9} T^{7} + p^{12} T^{8} \)
31$D_4\times C_2$ \( 1 - 8 p T + 21538 T^{2} + 156928 p T^{3} - 1122488189 T^{4} + 156928 p^{4} T^{5} + 21538 p^{6} T^{6} - 8 p^{10} T^{7} + p^{12} T^{8} \)
37$D_{4}$ \( ( 1 + 434 T + 148287 T^{2} + 434 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
41$D_4\times C_2$ \( 1 - 192 T - 57922 T^{2} + 8266752 T^{3} + 1693577811 T^{4} + 8266752 p^{3} T^{5} - 57922 p^{6} T^{6} - 192 p^{9} T^{7} + p^{12} T^{8} \)
43$D_4\times C_2$ \( 1 - 608 T + 119206 T^{2} - 55597952 T^{3} + 27016971067 T^{4} - 55597952 p^{3} T^{5} + 119206 p^{6} T^{6} - 608 p^{9} T^{7} + p^{12} T^{8} \)
47$D_4\times C_2$ \( 1 + 384 T - 62062 T^{2} + 718848 T^{3} + 17809252707 T^{4} + 718848 p^{3} T^{5} - 62062 p^{6} T^{6} + 384 p^{9} T^{7} + p^{12} T^{8} \)
53$D_{4}$ \( ( 1 - 408 T + 183418 T^{2} - 408 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
59$D_4\times C_2$ \( 1 + 1008 T + 378938 T^{2} + 228178944 T^{3} + 155070797979 T^{4} + 228178944 p^{3} T^{5} + 378938 p^{6} T^{6} + 1008 p^{9} T^{7} + p^{12} T^{8} \)
61$D_4\times C_2$ \( 1 + 742 T - 32291 T^{2} + 1567846 p T^{3} + 43867780 p^{2} T^{4} + 1567846 p^{4} T^{5} - 32291 p^{6} T^{6} + 742 p^{9} T^{7} + p^{12} T^{8} \)
67$D_4\times C_2$ \( 1 - 104 T - 79226 T^{2} + 53194336 T^{3} - 85603873013 T^{4} + 53194336 p^{3} T^{5} - 79226 p^{6} T^{6} - 104 p^{9} T^{7} + p^{12} T^{8} \)
71$D_{4}$ \( ( 1 - 1140 T + 1038994 T^{2} - 1140 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
73$D_{4}$ \( ( 1 - 850 T + 709827 T^{2} - 850 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
79$D_4\times C_2$ \( 1 - 440 T - 773378 T^{2} + 8404000 T^{3} + 596506646563 T^{4} + 8404000 p^{3} T^{5} - 773378 p^{6} T^{6} - 440 p^{9} T^{7} + p^{12} T^{8} \)
83$D_4\times C_2$ \( 1 - 264 T - 994102 T^{2} + 21060864 T^{3} + 764874544683 T^{4} + 21060864 p^{3} T^{5} - 994102 p^{6} T^{6} - 264 p^{9} T^{7} + p^{12} T^{8} \)
89$D_{4}$ \( ( 1 + 768 T + 1343527 T^{2} + 768 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 - 764 T - 1302902 T^{2} - 46796528 T^{3} + 2193556429267 T^{4} - 46796528 p^{3} T^{5} - 1302902 p^{6} T^{6} - 764 p^{9} T^{7} + p^{12} T^{8} \)
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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

−8.998368759341429306530200994417, −8.803316782762476577126355291837, −8.726353430105103288298105878155, −8.105387781285197801906498311118, −8.026126270263768653965562311445, −7.12960693755106879918702996602, −6.92262715704729926057755687537, −6.82597914966550234224095616781, −6.65619446453964322406335967405, −6.45398174561366671329717195593, −6.12623507871714322945051037723, −5.72606297497477389862627349745, −5.37497408307283654403195497968, −4.96456807204311572963434566352, −4.70011602710937871464307076658, −4.65601391033279153591822417531, −4.12272871773412943838636634483, −3.93231842492089854172311952094, −3.19432041675979517994047283245, −3.10525931143210135911954977396, −2.66798676908221348239208489808, −2.08149992295496464539812355347, −1.99834062961252519655337517313, −0.74933473418558299744814902238, −0.69839807334266746232937105452, 0.69839807334266746232937105452, 0.74933473418558299744814902238, 1.99834062961252519655337517313, 2.08149992295496464539812355347, 2.66798676908221348239208489808, 3.10525931143210135911954977396, 3.19432041675979517994047283245, 3.93231842492089854172311952094, 4.12272871773412943838636634483, 4.65601391033279153591822417531, 4.70011602710937871464307076658, 4.96456807204311572963434566352, 5.37497408307283654403195497968, 5.72606297497477389862627349745, 6.12623507871714322945051037723, 6.45398174561366671329717195593, 6.65619446453964322406335967405, 6.82597914966550234224095616781, 6.92262715704729926057755687537, 7.12960693755106879918702996602, 8.026126270263768653965562311445, 8.105387781285197801906498311118, 8.726353430105103288298105878155, 8.803316782762476577126355291837, 8.998368759341429306530200994417

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