Properties

Label 8-3630e4-1.1-c1e4-0-0
Degree $8$
Conductor $1.736\times 10^{14}$
Sign $1$
Analytic cond. $705886.$
Root an. cond. $5.38383$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s + 4·3-s + 10·4-s − 4·5-s − 16·6-s − 7-s − 20·8-s + 10·9-s + 16·10-s + 40·12-s − 2·13-s + 4·14-s − 16·15-s + 35·16-s − 11·17-s − 40·18-s − 19-s − 40·20-s − 4·21-s − 80·24-s + 10·25-s + 8·26-s + 20·27-s − 10·28-s − 9·29-s + 64·30-s + 17·31-s + ⋯
L(s)  = 1  − 2.82·2-s + 2.30·3-s + 5·4-s − 1.78·5-s − 6.53·6-s − 0.377·7-s − 7.07·8-s + 10/3·9-s + 5.05·10-s + 11.5·12-s − 0.554·13-s + 1.06·14-s − 4.13·15-s + 35/4·16-s − 2.66·17-s − 9.42·18-s − 0.229·19-s − 8.94·20-s − 0.872·21-s − 16.3·24-s + 2·25-s + 1.56·26-s + 3.84·27-s − 1.88·28-s − 1.67·29-s + 11.6·30-s + 3.05·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 11^{8}\)
Sign: $1$
Analytic conductor: \(705886.\)
Root analytic conductor: \(5.38383\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 11^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.283818094\)
\(L(\frac12)\) \(\approx\) \(2.283818094\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{4} \)
3$C_1$ \( ( 1 - T )^{4} \)
5$C_1$ \( ( 1 + T )^{4} \)
11 \( 1 \)
good7$(((C_4 \times C_2): C_2):C_2):C_2$ \( 1 + T - T^{2} - 13 T^{3} + 4 T^{4} - 13 p T^{5} - p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 + 2 T + 11 T^{2} + 36 T^{3} + 69 T^{4} + 36 p T^{5} + 11 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 + 11 T + 89 T^{2} + 517 T^{3} + 2464 T^{4} + 517 p T^{5} + 89 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 + T + 37 T^{2} + 3 T^{3} + 680 T^{4} + 3 p T^{5} + 37 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 + 27 T^{2} + 130 T^{3} + 239 T^{4} + 130 p T^{5} + 27 p^{2} T^{6} + p^{4} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 + 9 T + 67 T^{2} + 297 T^{3} + 1480 T^{4} + 297 p T^{5} + 67 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 - 17 T + 203 T^{2} - 1619 T^{3} + 10500 T^{4} - 1619 p T^{5} + 203 p^{2} T^{6} - 17 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 - 8 T + 77 T^{2} - 540 T^{3} + 4701 T^{4} - 540 p T^{5} + 77 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 - 11 T + 35 T^{2} + 201 T^{3} - 56 p T^{4} + 201 p T^{5} + 35 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 + T + 133 T^{2} + 75 T^{3} + 7736 T^{4} + 75 p T^{5} + 133 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 - 10 T + 173 T^{2} - 1050 T^{3} + 11099 T^{4} - 1050 p T^{5} + 173 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 - 11 T + 213 T^{2} - 1685 T^{3} + 16916 T^{4} - 1685 p T^{5} + 213 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \)
59$D_{4}$ \( ( 1 - 5 T + 63 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \)
61$C_2 \wr C_2\wr C_2$ \( 1 - 10 T + 144 T^{2} - 1670 T^{3} + 11406 T^{4} - 1670 p T^{5} + 144 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 - 9 T + 179 T^{2} - 1883 T^{3} + 15084 T^{4} - 1883 p T^{5} + 179 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 - 10 T + 224 T^{2} - 1650 T^{3} + 23006 T^{4} - 1650 p T^{5} + 224 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
73$D_{4}$ \( ( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
79$C_2 \wr C_2\wr C_2$ \( 1 - 7 T + 115 T^{2} - 727 T^{3} + 14764 T^{4} - 727 p T^{5} + 115 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2$ \( ( 1 + 8 T + p T^{2} )^{4} \)
89$C_2 \wr C_2\wr C_2$ \( 1 + 23 T + 435 T^{2} + 4963 T^{3} + 54864 T^{4} + 4963 p T^{5} + 435 p^{2} T^{6} + 23 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 + 2 T + 252 T^{2} + 910 T^{3} + 31206 T^{4} + 910 p T^{5} + 252 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−6.40898841898195952933739862030, −5.92232457284215921690748596301, −5.80139436279987031756787638290, −5.56599254653232204600744508576, −5.36911321368511256568053467372, −4.77923635028222352722515884064, −4.75715458493433087119016053323, −4.38194391837689542676467731161, −4.37512268286373812105764396224, −4.01522960581027692485207500016, −3.87923147939137407923006163578, −3.84424762418073461148210403134, −3.49807433351165353383522836487, −3.02710686435617900091243405588, −2.82621497415763705522349213394, −2.78665342128887718363509184340, −2.70436819100496471073299212197, −2.26725531144623771366865949323, −2.07438332108326373170591494564, −1.95190734589808946664458315961, −1.80011881606315677455541622499, −1.02231059918694922923337638275, −0.846300699405874323172181385590, −0.58796458724728536543363997911, −0.47272561813713079775061523352, 0.47272561813713079775061523352, 0.58796458724728536543363997911, 0.846300699405874323172181385590, 1.02231059918694922923337638275, 1.80011881606315677455541622499, 1.95190734589808946664458315961, 2.07438332108326373170591494564, 2.26725531144623771366865949323, 2.70436819100496471073299212197, 2.78665342128887718363509184340, 2.82621497415763705522349213394, 3.02710686435617900091243405588, 3.49807433351165353383522836487, 3.84424762418073461148210403134, 3.87923147939137407923006163578, 4.01522960581027692485207500016, 4.37512268286373812105764396224, 4.38194391837689542676467731161, 4.75715458493433087119016053323, 4.77923635028222352722515884064, 5.36911321368511256568053467372, 5.56599254653232204600744508576, 5.80139436279987031756787638290, 5.92232457284215921690748596301, 6.40898841898195952933739862030

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