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Properties

Label 8-3312e4-1.1-c1e4-0-1
Degree $8$
Conductor $1.203\times 10^{14}$
Sign $1$
Analytic cond. $489181.$
Root an. cond. $5.14261$
Motivic weight $1$
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  + 4·5-s − 2·7-s − 2·11-s + 4·13-s + 2·17-s − 8·19-s + 4·23-s + 4·25-s + 12·29-s − 12·31-s − 8·35-s + 14·37-s + 4·41-s − 4·43-s + 12·47-s + 2·49-s + 8·53-s − 8·55-s + 16·59-s + 6·61-s + 16·65-s − 8·67-s + 12·71-s + 16·73-s + 4·77-s − 10·79-s + 14·83-s + ⋯
L(s)  = 1  + 1.78·5-s − 0.755·7-s − 0.603·11-s + 1.10·13-s + 0.485·17-s − 1.83·19-s + 0.834·23-s + 4/5·25-s + 2.22·29-s − 2.15·31-s − 1.35·35-s + 2.30·37-s + 0.624·41-s − 0.609·43-s + 1.75·47-s + 2/7·49-s + 1.09·53-s − 1.07·55-s + 2.08·59-s + 0.768·61-s + 1.98·65-s − 0.977·67-s + 1.42·71-s + 1.87·73-s + 0.455·77-s − 1.12·79-s + 1.53·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 3^{8} \cdot 23^{4}\)
Sign: $1$
Analytic conductor: \(489181.\)
Root analytic conductor: \(5.14261\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 3^{8} \cdot 23^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(6.340255667\)
\(L(\frac12)\) \(\approx\) \(6.340255667\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3 \( 1 \)
23$C_1$ \( ( 1 - T )^{4} \)
good5$D_{4}$ \( ( 1 - 2 T + 4 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
7$(C_4^2 : C_2):C_2$ \( 1 + 2 T + 2 T^{2} - 6 T^{3} + 2 T^{4} - 6 p T^{5} + 2 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
11$C_2 \wr C_2\wr C_2$ \( 1 + 2 T + 2 p T^{2} - 6 T^{3} + 194 T^{4} - 6 p T^{5} + 2 p^{3} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 32 T^{2} - 108 T^{3} + 526 T^{4} - 108 p T^{5} + 32 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 46 T^{2} - 30 T^{3} + 938 T^{4} - 30 p T^{5} + 46 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 60 T^{2} + 216 T^{3} + 1114 T^{4} + 216 p T^{5} + 60 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 - 12 T + 144 T^{2} - 996 T^{3} + 6574 T^{4} - 996 p T^{5} + 144 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 + 12 T + 152 T^{2} + 1068 T^{3} + 7406 T^{4} + 1068 p T^{5} + 152 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 - 14 T + 138 T^{2} - 938 T^{3} + 5882 T^{4} - 938 p T^{5} + 138 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 88 T^{2} - 108 T^{3} + 3662 T^{4} - 108 p T^{5} + 88 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 28 T^{2} + 220 T^{3} + 3178 T^{4} + 220 p T^{5} + 28 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 - 12 T + 132 T^{2} - 972 T^{3} + 8006 T^{4} - 972 p T^{5} + 132 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 - 8 T + 140 T^{2} - 920 T^{3} + 10906 T^{4} - 920 p T^{5} + 140 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 - 16 T + 144 T^{2} - 912 T^{3} + 5166 T^{4} - 912 p T^{5} + 144 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 - 6 T + 234 T^{2} - 1090 T^{3} + 21114 T^{4} - 1090 p T^{5} + 234 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 140 T^{2} + 696 T^{3} + 10906 T^{4} + 696 p T^{5} + 140 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 - 12 T + 256 T^{2} - 2396 T^{3} + 26398 T^{4} - 2396 p T^{5} + 256 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 - 16 T + 224 T^{2} - 2112 T^{3} + 21950 T^{4} - 2112 p T^{5} + 224 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 + 10 T + 282 T^{2} + 2082 T^{3} + 32722 T^{4} + 2082 p T^{5} + 282 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 - 14 T + 382 T^{2} - 3542 T^{3} + 49650 T^{4} - 3542 p T^{5} + 382 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 - 14 T + 230 T^{2} - 2954 T^{3} + 28234 T^{4} - 2954 p T^{5} + 230 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 284 T^{2} - 1116 T^{3} + 37766 T^{4} - 1116 p T^{5} + 284 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
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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

−6.28015954078148862630983594066, −5.89209135323419842232442809128, −5.69754992257747941192472380377, −5.56300262116029896622615993238, −5.32979695609213789300019574835, −5.21584648345157887912763404869, −5.10991376729094436419768349119, −4.59664950795368886719792791491, −4.53653191890917402206186955000, −4.21392792790787580774487762792, −4.02611128600815949456764041967, −3.93095555862608866189639992860, −3.60575444108104971944660674066, −3.29706013685581951883348985007, −3.14919193884367597004956209014, −2.95640677305405847776877680543, −2.45034783982930936247893157199, −2.35556987045244911599982836097, −2.25864396690449414035366064835, −1.98792075253588630927137414199, −1.87858449954833478737123682163, −1.11685231277491393902089433064, −1.00196158267255976486742537928, −0.880996477802219952126540561766, −0.35710540578101969505754034865, 0.35710540578101969505754034865, 0.880996477802219952126540561766, 1.00196158267255976486742537928, 1.11685231277491393902089433064, 1.87858449954833478737123682163, 1.98792075253588630927137414199, 2.25864396690449414035366064835, 2.35556987045244911599982836097, 2.45034783982930936247893157199, 2.95640677305405847776877680543, 3.14919193884367597004956209014, 3.29706013685581951883348985007, 3.60575444108104971944660674066, 3.93095555862608866189639992860, 4.02611128600815949456764041967, 4.21392792790787580774487762792, 4.53653191890917402206186955000, 4.59664950795368886719792791491, 5.10991376729094436419768349119, 5.21584648345157887912763404869, 5.32979695609213789300019574835, 5.56300262116029896622615993238, 5.69754992257747941192472380377, 5.89209135323419842232442809128, 6.28015954078148862630983594066

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