Properties

Label 8-24e8-1.1-c1e4-0-9
Degree $8$
Conductor $110075314176$
Sign $1$
Analytic cond. $447.505$
Root an. cond. $2.14461$
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  + 3-s − 5-s + 3·7-s + 3·9-s − 2·11-s + 5·13-s − 15-s − 10·17-s + 14·19-s + 3·21-s + 5·23-s + 2·25-s + 8·27-s − 3·29-s + 7·31-s − 2·33-s − 3·35-s − 12·37-s + 5·39-s + 12·41-s − 8·43-s − 3·45-s + 3·47-s + 8·49-s − 10·51-s + 20·53-s + 2·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 1.13·7-s + 9-s − 0.603·11-s + 1.38·13-s − 0.258·15-s − 2.42·17-s + 3.21·19-s + 0.654·21-s + 1.04·23-s + 2/5·25-s + 1.53·27-s − 0.557·29-s + 1.25·31-s − 0.348·33-s − 0.507·35-s − 1.97·37-s + 0.800·39-s + 1.87·41-s − 1.21·43-s − 0.447·45-s + 0.437·47-s + 8/7·49-s − 1.40·51-s + 2.74·53-s + 0.269·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{24} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(447.505\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{24} \cdot 3^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(4.654572109\)
\(L(\frac12)\) \(\approx\) \(4.654572109\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \)
good5$D_4\times C_2$ \( 1 + T - T^{2} - 8 T^{3} - 26 T^{4} - 8 p T^{5} - p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
7$D_4\times C_2$ \( 1 - 3 T + T^{2} + 18 T^{3} - 48 T^{4} + 18 p T^{5} + p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
11$C_2^2$ \( ( 1 + T - 10 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \)
13$D_4\times C_2$ \( 1 - 5 T + T^{2} + 10 T^{3} + 82 T^{4} + 10 p T^{5} + p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \)
17$D_{4}$ \( ( 1 + 5 T + 32 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \)
19$D_{4}$ \( ( 1 - 7 T + 42 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 - 5 T - 19 T^{2} + 10 T^{3} + 832 T^{4} + 10 p T^{5} - 19 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \)
29$D_4\times C_2$ \( 1 + 3 T - 43 T^{2} - 18 T^{3} + 1602 T^{4} - 18 p T^{5} - 43 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
31$D_4\times C_2$ \( 1 - 7 T - 17 T^{2} - 28 T^{3} + 1876 T^{4} - 28 p T^{5} - 17 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \)
37$D_{4}$ \( ( 1 + 6 T + 50 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
41$D_4\times C_2$ \( 1 - 12 T + 59 T^{2} - 36 T^{3} - 360 T^{4} - 36 p T^{5} + 59 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
43$D_4\times C_2$ \( 1 + 8 T - 5 T^{2} - 136 T^{3} + 160 T^{4} - 136 p T^{5} - 5 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
47$D_4\times C_2$ \( 1 - 3 T - 79 T^{2} + 18 T^{3} + 5112 T^{4} + 18 p T^{5} - 79 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
53$D_{4}$ \( ( 1 - 10 T + 98 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + 7 T - 10 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \)
61$D_4\times C_2$ \( 1 + T - 113 T^{2} - 8 T^{3} + 9214 T^{4} - 8 p T^{5} - 113 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
67$D_4\times C_2$ \( 1 + 4 T - 89 T^{2} - 116 T^{3} + 5464 T^{4} - 116 p T^{5} - 89 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2$ \( ( 1 - 4 T + p T^{2} )^{4} \)
73$D_{4}$ \( ( 1 + 7 T + 84 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \)
79$D_4\times C_2$ \( 1 + 7 T - 113 T^{2} + 28 T^{3} + 16132 T^{4} + 28 p T^{5} - 113 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \)
83$D_4\times C_2$ \( 1 + 25 T + 311 T^{2} + 3700 T^{3} + 39832 T^{4} + 3700 p T^{5} + 311 p^{2} T^{6} + 25 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{4} \)
97$D_4\times C_2$ \( 1 - 8 T - 113 T^{2} + 136 T^{3} + 15712 T^{4} + 136 p T^{5} - 113 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
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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

−7.60404459134309995703150582323, −7.46557633662228301988868131620, −7.28308354178501013842827571022, −7.09759605788568593513571401046, −7.07559063708724192077585617480, −6.61851754281682846513484341495, −6.19408249706713889815748821842, −6.07374990211106647349687793982, −5.97254022205530949569961212358, −5.23078256214466705528109155561, −5.13686728466823317168480135780, −5.11289102859174306019458893538, −4.86640272009447604577734518458, −4.40636548884894184863420455169, −4.22826253247854884092107840023, −3.89823593214760002329915209241, −3.64989229300377999016487269279, −3.41624465900286287376537845126, −2.77596015354988714702831995339, −2.72255292496608919843230168757, −2.59305142203443513505009289853, −1.79180421080128494628330402398, −1.54026576270405404934240011293, −1.16474554035577397116627094911, −0.71833920817676980901003929952, 0.71833920817676980901003929952, 1.16474554035577397116627094911, 1.54026576270405404934240011293, 1.79180421080128494628330402398, 2.59305142203443513505009289853, 2.72255292496608919843230168757, 2.77596015354988714702831995339, 3.41624465900286287376537845126, 3.64989229300377999016487269279, 3.89823593214760002329915209241, 4.22826253247854884092107840023, 4.40636548884894184863420455169, 4.86640272009447604577734518458, 5.11289102859174306019458893538, 5.13686728466823317168480135780, 5.23078256214466705528109155561, 5.97254022205530949569961212358, 6.07374990211106647349687793982, 6.19408249706713889815748821842, 6.61851754281682846513484341495, 7.07559063708724192077585617480, 7.09759605788568593513571401046, 7.28308354178501013842827571022, 7.46557633662228301988868131620, 7.60404459134309995703150582323

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