Properties

Label 8-1200e4-1.1-c2e4-0-16
Degree $8$
Conductor $2.074\times 10^{12}$
Sign $1$
Analytic cond. $1.14304\times 10^{6}$
Root an. cond. $5.71818$
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  − 6·9-s − 4·13-s + 48·17-s − 96·29-s + 88·37-s − 24·41-s − 74·49-s − 120·53-s − 148·61-s − 280·73-s + 27·81-s − 384·89-s + 140·97-s − 456·101-s − 380·109-s − 432·113-s + 24·117-s + 220·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 288·153-s + 157-s + 163-s + ⋯
L(s)  = 1  − 2/3·9-s − 0.307·13-s + 2.82·17-s − 3.31·29-s + 2.37·37-s − 0.585·41-s − 1.51·49-s − 2.26·53-s − 2.42·61-s − 3.83·73-s + 1/3·81-s − 4.31·89-s + 1.44·97-s − 4.51·101-s − 3.48·109-s − 3.82·113-s + 8/39·117-s + 1.81·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s − 1.88·153-s + 0.00636·157-s + 0.00613·163-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 3^{4} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(1.14304\times 10^{6}\)
Root analytic conductor: \(5.71818\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1200} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 3^{4} \cdot 5^{8} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.9511234954\)
\(L(\frac12)\) \(\approx\) \(0.9511234954\)
\(L(2)\) 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 \( 1 \)
3$C_2$ \( ( 1 + p T^{2} )^{2} \)
5 \( 1 \)
good7$D_4\times C_2$ \( 1 + 74 T^{2} + 4587 T^{4} + 74 p^{4} T^{6} + p^{8} T^{8} \)
11$C_2^2$ \( ( 1 - 10 p T^{2} + p^{4} T^{4} )^{2} \)
13$C_2$ \( ( 1 + T + p^{2} T^{2} )^{4} \)
17$D_{4}$ \( ( 1 - 24 T + 326 T^{2} - 24 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
19$D_4\times C_2$ \( 1 - 1174 T^{2} + 603627 T^{4} - 1174 p^{4} T^{6} + p^{8} T^{8} \)
23$D_4\times C_2$ \( 1 - 988 T^{2} + 575622 T^{4} - 988 p^{4} T^{6} + p^{8} T^{8} \)
29$D_{4}$ \( ( 1 + 48 T + 1862 T^{2} + 48 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
31$D_4\times C_2$ \( 1 - 2230 T^{2} + 2733867 T^{4} - 2230 p^{4} T^{6} + p^{8} T^{8} \)
37$D_{4}$ \( ( 1 - 44 T + 1638 T^{2} - 44 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
41$D_{4}$ \( ( 1 + 12 T - 166 T^{2} + 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - 4486 T^{2} + 11170107 T^{4} - 4486 p^{4} T^{6} + p^{8} T^{8} \)
47$D_4\times C_2$ \( 1 - 4396 T^{2} + 13678182 T^{4} - 4396 p^{4} T^{6} + p^{8} T^{8} \)
53$D_{4}$ \( ( 1 + 60 T + 2954 T^{2} + 60 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
59$D_4\times C_2$ \( 1 - 4300 T^{2} + 6047622 T^{4} - 4300 p^{4} T^{6} + p^{8} T^{8} \)
61$D_{4}$ \( ( 1 + 74 T + 7227 T^{2} + 74 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 - 4486 T^{2} + 18973947 T^{4} - 4486 p^{4} T^{6} + p^{8} T^{8} \)
71$D_4\times C_2$ \( 1 - 19684 T^{2} + 147631302 T^{4} - 19684 p^{4} T^{6} + p^{8} T^{8} \)
73$D_{4}$ \( ( 1 + 140 T + 13974 T^{2} + 140 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
79$D_4\times C_2$ \( 1 - 16036 T^{2} + 122318790 T^{4} - 16036 p^{4} T^{6} + p^{8} T^{8} \)
83$C_2^2$ \( ( 1 - 8486 T^{2} + p^{4} T^{4} )^{2} \)
89$C_2$ \( ( 1 + 96 T + p^{2} T^{2} )^{4} \)
97$D_{4}$ \( ( 1 - 70 T + 13707 T^{2} - 70 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
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

−6.78707475930119520724352853981, −6.58694404019788779390719963368, −6.17715953338077159429224321614, −6.00988460806797030920317303831, −5.84639879197809399735045669163, −5.63735093767885717810741840816, −5.56108954670236944859400498337, −5.24730771505535180515753459078, −5.07035891400129602852457725136, −4.75044621613972284100009956749, −4.48504213342667199871517589658, −4.06570935784412246583805422079, −4.04539048718092034654886156116, −3.79734119896161384553926153180, −3.48136654960574363727929517336, −2.97943278968808529750653695261, −2.87385451524246505258419061156, −2.82448495822063573100068480623, −2.71150164229717066424659570281, −1.81920875584004262192883270594, −1.53312960317769268779095413611, −1.46062123391020292710102013013, −1.38299771700804754031691973468, −0.47116879297640367747044423463, −0.18595417083415574571622828042, 0.18595417083415574571622828042, 0.47116879297640367747044423463, 1.38299771700804754031691973468, 1.46062123391020292710102013013, 1.53312960317769268779095413611, 1.81920875584004262192883270594, 2.71150164229717066424659570281, 2.82448495822063573100068480623, 2.87385451524246505258419061156, 2.97943278968808529750653695261, 3.48136654960574363727929517336, 3.79734119896161384553926153180, 4.04539048718092034654886156116, 4.06570935784412246583805422079, 4.48504213342667199871517589658, 4.75044621613972284100009956749, 5.07035891400129602852457725136, 5.24730771505535180515753459078, 5.56108954670236944859400498337, 5.63735093767885717810741840816, 5.84639879197809399735045669163, 6.00988460806797030920317303831, 6.17715953338077159429224321614, 6.58694404019788779390719963368, 6.78707475930119520724352853981

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