Properties

Label 8-42e4-1.1-c2e4-0-0
Degree $8$
Conductor $3111696$
Sign $1$
Analytic cond. $1.71528$
Root an. cond. $1.06977$
Motivic weight $2$
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  − 4·4-s + 10·9-s + 40·13-s + 12·16-s − 64·19-s + 56·25-s − 128·31-s − 40·36-s + 80·37-s + 80·43-s + 14·49-s − 160·52-s − 56·61-s − 32·64-s + 240·67-s + 120·73-s + 256·76-s − 128·79-s + 19·81-s − 360·97-s − 224·100-s − 160·103-s − 72·109-s + 400·117-s + 272·121-s + 512·124-s + 127-s + ⋯
L(s)  = 1  − 4-s + 10/9·9-s + 3.07·13-s + 3/4·16-s − 3.36·19-s + 2.23·25-s − 4.12·31-s − 1.11·36-s + 2.16·37-s + 1.86·43-s + 2/7·49-s − 3.07·52-s − 0.918·61-s − 1/2·64-s + 3.58·67-s + 1.64·73-s + 3.36·76-s − 1.62·79-s + 0.234·81-s − 3.71·97-s − 2.23·100-s − 1.55·103-s − 0.660·109-s + 3.41·117-s + 2.24·121-s + 4.12·124-s + 0.00787·127-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3111696\)    =    \(2^{4} \cdot 3^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(1.71528\)
Root analytic conductor: \(1.06977\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3111696,\ (\ :1, 1, 1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.104426905\)
\(L(\frac12)\) \(\approx\) \(1.104426905\)
\(L(2)\) 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$C_2$ \( ( 1 + p T^{2} )^{2} \)
3$C_2^2$ \( 1 - 10 T^{2} + p^{4} T^{4} \)
7$C_2$ \( ( 1 - p T^{2} )^{2} \)
good5$D_4\times C_2$ \( 1 - 56 T^{2} + 1586 T^{4} - 56 p^{4} T^{6} + p^{8} T^{8} \)
11$D_4\times C_2$ \( 1 - 272 T^{2} + 36578 T^{4} - 272 p^{4} T^{6} + p^{8} T^{8} \)
13$D_{4}$ \( ( 1 - 20 T + 326 T^{2} - 20 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
17$D_4\times C_2$ \( 1 - 440 T^{2} + 204242 T^{4} - 440 p^{4} T^{6} + p^{8} T^{8} \)
19$C_2$ \( ( 1 + 16 T + p^{2} T^{2} )^{4} \)
23$D_4\times C_2$ \( 1 + 688 T^{2} + 666818 T^{4} + 688 p^{4} T^{6} + p^{8} T^{8} \)
29$D_4\times C_2$ \( 1 - 1652 T^{2} + 1917638 T^{4} - 1652 p^{4} T^{6} + p^{8} T^{8} \)
31$D_{4}$ \( ( 1 + 64 T + 2246 T^{2} + 64 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
37$C_2$ \( ( 1 - 20 T + p^{2} T^{2} )^{4} \)
41$D_4\times C_2$ \( 1 - 856 T^{2} - 2330094 T^{4} - 856 p^{4} T^{6} + p^{8} T^{8} \)
43$D_{4}$ \( ( 1 - 40 T + 3090 T^{2} - 40 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 - 4346 T^{2} + p^{4} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - 3026 T^{2} + p^{4} T^{4} )^{2} \)
59$D_4\times C_2$ \( 1 - 10532 T^{2} + 49098278 T^{4} - 10532 p^{4} T^{6} + p^{8} T^{8} \)
61$D_{4}$ \( ( 1 + 28 T + 4838 T^{2} + 28 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
67$D_{4}$ \( ( 1 - 120 T + 12466 T^{2} - 120 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
71$D_4\times C_2$ \( 1 - 12176 T^{2} + 74167106 T^{4} - 12176 p^{4} T^{6} + p^{8} T^{8} \)
73$D_{4}$ \( ( 1 - 60 T + 9766 T^{2} - 60 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
79$D_{4}$ \( ( 1 + 64 T + 10706 T^{2} + 64 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - 6356 T^{2} - 6983674 T^{4} - 6356 p^{4} T^{6} + p^{8} T^{8} \)
89$D_4\times C_2$ \( 1 - 27832 T^{2} + 318232338 T^{4} - 27832 p^{4} T^{6} + p^{8} T^{8} \)
97$D_{4}$ \( ( 1 + 180 T + 26470 T^{2} + 180 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
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

−11.80190145350659534624340353683, −11.00940913991854171277319046275, −10.99922140489800898182747560996, −10.97656607736838537880749855765, −10.64329340130072194090857144775, −10.48806283517269266828015213592, −9.497228945272114838058054057530, −9.453655161565392657145870140473, −9.297629870369767403113554207709, −8.618024251648609891486181622236, −8.598351746523521937167876874304, −8.221416607500409401293273300572, −7.984828802260199342245516498849, −7.18679619517840313424465440146, −6.82977106379627389499134696531, −6.67563390324487413403602582116, −5.93492721119400911993870894085, −5.84403132523823845965565988871, −5.31119261484233062466682806038, −4.49681830592379270655001915085, −4.11464777194798800290221936895, −4.00088304407135574630311318648, −3.44301175405416403260060461427, −2.25442640976318514445653685193, −1.23197583297330044693123404601, 1.23197583297330044693123404601, 2.25442640976318514445653685193, 3.44301175405416403260060461427, 4.00088304407135574630311318648, 4.11464777194798800290221936895, 4.49681830592379270655001915085, 5.31119261484233062466682806038, 5.84403132523823845965565988871, 5.93492721119400911993870894085, 6.67563390324487413403602582116, 6.82977106379627389499134696531, 7.18679619517840313424465440146, 7.984828802260199342245516498849, 8.221416607500409401293273300572, 8.598351746523521937167876874304, 8.618024251648609891486181622236, 9.297629870369767403113554207709, 9.453655161565392657145870140473, 9.497228945272114838058054057530, 10.48806283517269266828015213592, 10.64329340130072194090857144775, 10.97656607736838537880749855765, 10.99922140489800898182747560996, 11.00940913991854171277319046275, 11.80190145350659534624340353683

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