Properties

Label 8-9680e4-1.1-c1e4-0-11
Degree $8$
Conductor $8.780\times 10^{15}$
Sign $1$
Analytic cond. $3.56952\times 10^{7}$
Root an. cond. $8.79176$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 4·5-s − 6·7-s − 2·9-s + 4·13-s − 8·15-s − 12·21-s − 4·23-s + 10·25-s − 8·27-s + 16·29-s − 16·31-s + 24·35-s − 8·37-s + 8·39-s + 8·41-s + 10·43-s + 8·45-s − 14·47-s + 2·49-s + 8·53-s − 4·61-s + 12·63-s − 16·65-s + 2·67-s − 8·69-s − 8·71-s + ⋯
L(s)  = 1  + 1.15·3-s − 1.78·5-s − 2.26·7-s − 2/3·9-s + 1.10·13-s − 2.06·15-s − 2.61·21-s − 0.834·23-s + 2·25-s − 1.53·27-s + 2.97·29-s − 2.87·31-s + 4.05·35-s − 1.31·37-s + 1.28·39-s + 1.24·41-s + 1.52·43-s + 1.19·45-s − 2.04·47-s + 2/7·49-s + 1.09·53-s − 0.512·61-s + 1.51·63-s − 1.98·65-s + 0.244·67-s − 0.963·69-s − 0.949·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 5^{4} \cdot 11^{8}\)
Sign: $1$
Analytic conductor: \(3.56952\times 10^{7}\)
Root analytic conductor: \(8.79176\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{16} \cdot 5^{4} \cdot 11^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_1$ \( ( 1 + T )^{4} \)
11 \( 1 \)
good3$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 2 p T^{2} - 8 T^{3} + 19 T^{4} - 8 p T^{5} + 2 p^{3} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
7$C_2 \wr C_2\wr C_2$ \( 1 + 6 T + 34 T^{2} + 120 T^{3} + 375 T^{4} + 120 p T^{5} + 34 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 40 T^{2} - 124 T^{3} + 718 T^{4} - 124 p T^{5} + 40 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2$ \( ( 1 + p T^{2} )^{4} \)
19$C_2 \wr C_2\wr C_2$ \( 1 + 40 T^{2} - 96 T^{3} + 750 T^{4} - 96 p T^{5} + 40 p^{2} T^{6} + p^{4} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 56 T^{2} + 292 T^{3} + 1534 T^{4} + 292 p T^{5} + 56 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2$ \( ( 1 - 4 T + p T^{2} )^{4} \)
31$C_2 \wr C_2\wr C_2$ \( 1 + 16 T + 160 T^{2} + 1120 T^{3} + 6862 T^{4} + 1120 p T^{5} + 160 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 100 T^{2} + 632 T^{3} + 4918 T^{4} + 632 p T^{5} + 100 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 - 8 T + 134 T^{2} - 992 T^{3} + 7627 T^{4} - 992 p T^{5} + 134 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 - 10 T + 118 T^{2} - 784 T^{3} + 6571 T^{4} - 784 p T^{5} + 118 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 + 14 T + 194 T^{2} + 1640 T^{3} + 13975 T^{4} + 1640 p T^{5} + 194 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 - 8 T + 200 T^{2} - 1256 T^{3} + 15598 T^{4} - 1256 p T^{5} + 200 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 + 128 T^{2} + 432 T^{3} + 7710 T^{4} + 432 p T^{5} + 128 p^{2} T^{6} + p^{4} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 154 T^{2} + 832 T^{3} + 11575 T^{4} + 832 p T^{5} + 154 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 166 T^{2} - 632 T^{3} + 13843 T^{4} - 632 p T^{5} + 166 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 128 T^{2} + 1496 T^{3} + 11758 T^{4} + 1496 p T^{5} + 128 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2$ \( ( 1 - 8 T + p T^{2} )^{4} \)
79$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 196 T^{2} - 292 T^{3} + 17734 T^{4} - 292 p T^{5} + 196 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 + 12 T + 152 T^{2} + 924 T^{3} + 6798 T^{4} + 924 p T^{5} + 152 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 - 12 T + 266 T^{2} - 1680 T^{3} + 27219 T^{4} - 1680 p T^{5} + 266 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 + 184 T^{2} + 528 T^{3} + 20286 T^{4} + 528 p T^{5} + 184 p^{2} T^{6} + p^{4} T^{8} \)
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

−5.94077380068457339052436533705, −5.44359917541206261409005861065, −5.29432827955211206983820061090, −5.22266987273651314427025890120, −5.19627108963083188116033047627, −4.73665502149639210799864936023, −4.60570777167381269508270313977, −4.28985920084398971814141238867, −4.13838515450513686723236955354, −3.91086508988740911223887757439, −3.88043908233057128780939454243, −3.61861444713474274083243338314, −3.57220705630795692261955713598, −3.30903414376991807038494748636, −3.19885794213389471721324091955, −2.92965251601321564998494036210, −2.85913156468548996137064698707, −2.65847384216052408046896124378, −2.40893930675955047539061388361, −2.21026382856769329858227086061, −2.00058167856979318063257307226, −1.55834490118308682450793088469, −1.16661977709532373427471207514, −1.08980022643092505653198845159, −0.935177326452036914089710265727, 0, 0, 0, 0, 0.935177326452036914089710265727, 1.08980022643092505653198845159, 1.16661977709532373427471207514, 1.55834490118308682450793088469, 2.00058167856979318063257307226, 2.21026382856769329858227086061, 2.40893930675955047539061388361, 2.65847384216052408046896124378, 2.85913156468548996137064698707, 2.92965251601321564998494036210, 3.19885794213389471721324091955, 3.30903414376991807038494748636, 3.57220705630795692261955713598, 3.61861444713474274083243338314, 3.88043908233057128780939454243, 3.91086508988740911223887757439, 4.13838515450513686723236955354, 4.28985920084398971814141238867, 4.60570777167381269508270313977, 4.73665502149639210799864936023, 5.19627108963083188116033047627, 5.22266987273651314427025890120, 5.29432827955211206983820061090, 5.44359917541206261409005861065, 5.94077380068457339052436533705

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