Properties

Label 8-1014e4-1.1-c1e4-0-4
Degree $8$
Conductor $1.057\times 10^{12}$
Sign $1$
Analytic cond. $4297.93$
Root an. cond. $2.84549$
Motivic weight $1$
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  − 2·3-s + 4-s − 6·7-s + 9-s + 6·11-s − 2·12-s − 8·17-s − 6·19-s + 12·21-s − 2·23-s + 6·25-s + 2·27-s − 6·28-s + 2·29-s − 12·33-s + 36-s + 12·37-s − 36·41-s − 2·43-s + 6·44-s + 8·49-s + 16·51-s − 12·53-s + 12·57-s + 8·61-s − 6·63-s − 64-s + ⋯
L(s)  = 1  − 1.15·3-s + 1/2·4-s − 2.26·7-s + 1/3·9-s + 1.80·11-s − 0.577·12-s − 1.94·17-s − 1.37·19-s + 2.61·21-s − 0.417·23-s + 6/5·25-s + 0.384·27-s − 1.13·28-s + 0.371·29-s − 2.08·33-s + 1/6·36-s + 1.97·37-s − 5.62·41-s − 0.304·43-s + 0.904·44-s + 8/7·49-s + 2.24·51-s − 1.64·53-s + 1.58·57-s + 1.02·61-s − 0.755·63-s − 1/8·64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 13^{8}\)
Sign: $1$
Analytic conductor: \(4297.93\)
Root analytic conductor: \(2.84549\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1014} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{4} \cdot 13^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.6619664070\)
\(L(\frac12)\) \(\approx\) \(0.6619664070\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - T^{2} + T^{4} \)
3$C_2$ \( ( 1 + T + T^{2} )^{2} \)
13 \( 1 \)
good5$C_2^2$$\times$$C_2^2$ \( ( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} )( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} ) \)
7$D_4\times C_2$ \( 1 + 6 T + 4 p T^{2} + 96 T^{3} + 291 T^{4} + 96 p T^{5} + 4 p^{3} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
11$D_4\times C_2$ \( 1 - 6 T + 28 T^{2} - 96 T^{3} + 267 T^{4} - 96 p T^{5} + 28 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2$$\times$$C_2^2$ \( ( 1 + 8 T + p T^{2} )^{2}( 1 - 8 T + 47 T^{2} - 8 p T^{3} + p^{2} T^{4} ) \)
19$D_4\times C_2$ \( 1 + 6 T + 44 T^{2} + 192 T^{3} + 891 T^{4} + 192 p T^{5} + 44 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
23$D_4\times C_2$ \( 1 + 2 T - 16 T^{2} - 52 T^{3} - 221 T^{4} - 52 p T^{5} - 16 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
29$D_4\times C_2$ \( 1 - 2 T - 43 T^{2} + 22 T^{3} + 1252 T^{4} + 22 p T^{5} - 43 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
31$D_4\times C_2$ \( 1 - 92 T^{2} + 3846 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} \)
37$D_4\times C_2$ \( 1 - 12 T + 85 T^{2} - 12 p T^{3} + 48 p T^{4} - 12 p^{2} T^{5} + 85 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
41$D_4\times C_2$ \( 1 + 36 T + 621 T^{2} + 6804 T^{3} + 51752 T^{4} + 6804 p T^{5} + 621 p^{2} T^{6} + 36 p^{3} T^{7} + p^{4} T^{8} \)
43$D_4\times C_2$ \( 1 + 2 T - 8 T^{2} - 148 T^{3} - 1877 T^{4} - 148 p T^{5} - 8 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
47$D_4\times C_2$ \( 1 - 116 T^{2} + 6810 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} \)
53$D_{4}$ \( ( 1 + 6 T + 103 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
59$C_2^3$ \( 1 + 54 T^{2} - 565 T^{4} + 54 p^{2} T^{6} + p^{4} T^{8} \)
61$D_4\times C_2$ \( 1 - 8 T - 47 T^{2} + 88 T^{3} + 4696 T^{4} + 88 p T^{5} - 47 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
67$D_4\times C_2$ \( 1 - 42 T + 868 T^{2} - 11760 T^{3} + 113307 T^{4} - 11760 p T^{5} + 868 p^{2} T^{6} - 42 p^{3} T^{7} + p^{4} T^{8} \)
71$D_4\times C_2$ \( 1 + 6 T + 148 T^{2} + 816 T^{3} + 14307 T^{4} + 816 p T^{5} + 148 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
73$D_4\times C_2$ \( 1 - 158 T^{2} + 16131 T^{4} - 158 p^{2} T^{6} + p^{4} T^{8} \)
79$D_{4}$ \( ( 1 + 12 T + 182 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - 228 T^{2} + 24074 T^{4} - 228 p^{2} T^{6} + p^{4} T^{8} \)
89$D_4\times C_2$ \( 1 - 12 T + 202 T^{2} - 1848 T^{3} + 20067 T^{4} - 1848 p T^{5} + 202 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2^3$ \( 1 + 158 T^{2} + 15555 T^{4} + 158 p^{2} T^{6} + p^{4} T^{8} \)
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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

−6.95342061251411452913824849256, −6.81489366428869604096207443635, −6.52008279617424131326157016970, −6.49069139393279802468953377967, −6.39283012958816740439915082328, −6.05946124444720685329282293368, −5.98973270845087938681972425065, −5.85331076841406113554900825368, −5.23399235460784412676336116036, −4.96930059717601724539577696471, −4.72896702418176111503571888232, −4.65214218448244023985792314115, −4.62181024477718259820167225721, −3.84559477538275105199807516812, −3.68702409204954292201059605349, −3.67662596668744703938691514945, −3.39410820782102921621289051727, −3.00471344361325431322676938020, −2.74254367422530931991557324841, −2.34213998501703105263335673941, −1.90140886989601695175632634054, −1.88200274704251013697720762676, −1.30135589303490913090778949103, −0.61935289763040152894160624879, −0.30343879879023789325653856103, 0.30343879879023789325653856103, 0.61935289763040152894160624879, 1.30135589303490913090778949103, 1.88200274704251013697720762676, 1.90140886989601695175632634054, 2.34213998501703105263335673941, 2.74254367422530931991557324841, 3.00471344361325431322676938020, 3.39410820782102921621289051727, 3.67662596668744703938691514945, 3.68702409204954292201059605349, 3.84559477538275105199807516812, 4.62181024477718259820167225721, 4.65214218448244023985792314115, 4.72896702418176111503571888232, 4.96930059717601724539577696471, 5.23399235460784412676336116036, 5.85331076841406113554900825368, 5.98973270845087938681972425065, 6.05946124444720685329282293368, 6.39283012958816740439915082328, 6.49069139393279802468953377967, 6.52008279617424131326157016970, 6.81489366428869604096207443635, 6.95342061251411452913824849256

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