Properties

Label 8-1323e4-1.1-c1e4-0-3
Degree $8$
Conductor $3.064\times 10^{12}$
Sign $1$
Analytic cond. $12455.1$
Root an. cond. $3.25026$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s + 8·5-s + 20·17-s − 16·20-s + 30·25-s + 8·37-s + 8·41-s + 12·47-s + 20·59-s − 36·67-s − 40·68-s − 28·79-s + 8·83-s + 160·85-s + 20·89-s − 60·100-s + 28·101-s + 48·109-s − 30·121-s + 80·125-s + 127-s + 131-s + 137-s + 139-s − 16·148-s + 149-s + 151-s + ⋯
L(s)  = 1  − 4-s + 3.57·5-s + 4.85·17-s − 3.57·20-s + 6·25-s + 1.31·37-s + 1.24·41-s + 1.75·47-s + 2.60·59-s − 4.39·67-s − 4.85·68-s − 3.15·79-s + 0.878·83-s + 17.3·85-s + 2.11·89-s − 6·100-s + 2.78·101-s + 4.59·109-s − 2.72·121-s + 7.15·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1.31·148-s + 0.0819·149-s + 0.0813·151-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{12} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(12455.1\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1323} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{12} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(10.89722014\)
\(L(\frac12)\) \(\approx\) \(10.89722014\)
\(L(\frac{3}{2})\) 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)$
bad3 \( 1 \)
7 \( 1 \)
good2$C_4\times C_2$ \( 1 + p T^{2} + p^{2} T^{4} + p^{3} T^{6} + p^{4} T^{8} \)
5$D_{4}$ \( ( 1 - 4 T + 9 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
11$D_4\times C_2$ \( 1 + 30 T^{2} + 422 T^{4} + 30 p^{2} T^{6} + p^{4} T^{8} \)
13$D_4\times C_2$ \( 1 + 46 T^{2} + 862 T^{4} + 46 p^{2} T^{6} + p^{4} T^{8} \)
17$C_2$ \( ( 1 - 5 T + p T^{2} )^{4} \)
19$D_4\times C_2$ \( 1 + 30 T^{2} + 902 T^{4} + 30 p^{2} T^{6} + p^{4} T^{8} \)
23$D_4\times C_2$ \( 1 + 56 T^{2} + 1522 T^{4} + 56 p^{2} T^{6} + p^{4} T^{8} \)
29$D_4\times C_2$ \( 1 - 10 T^{2} + 1582 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} \)
31$D_4\times C_2$ \( 1 + 70 T^{2} + 2902 T^{4} + 70 p^{2} T^{6} + p^{4} T^{8} \)
37$D_{4}$ \( ( 1 - 4 T + 33 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
41$D_{4}$ \( ( 1 - 4 T + p T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 + 81 T^{2} + p^{2} T^{4} )^{2} \)
47$D_{4}$ \( ( 1 - 6 T + 83 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 + 96 T^{2} + p^{2} T^{4} )^{2} \)
59$D_{4}$ \( ( 1 - 10 T + 63 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \)
61$D_4\times C_2$ \( 1 + 230 T^{2} + 20622 T^{4} + 230 p^{2} T^{6} + p^{4} T^{8} \)
67$D_{4}$ \( ( 1 + 18 T + 210 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2} \)
71$D_4\times C_2$ \( 1 + 68 T^{2} + 7318 T^{4} + 68 p^{2} T^{6} + p^{4} T^{8} \)
73$D_4\times C_2$ \( 1 + 112 T^{2} + 5794 T^{4} + 112 p^{2} T^{6} + p^{4} T^{8} \)
79$D_{4}$ \( ( 1 + 14 T + 187 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \)
83$D_{4}$ \( ( 1 - 4 T + 165 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
89$D_{4}$ \( ( 1 - 10 T + 158 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 + 358 T^{2} + 50734 T^{4} + 358 p^{2} T^{6} + p^{4} T^{8} \)
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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

−6.88769986878367372276323841186, −6.34505159661455297220667457763, −6.30223077681206188071927542378, −6.01599573298195131709137351758, −5.95935520872894079452128862703, −5.76352478425873227598209271367, −5.49076877856289621015410733327, −5.47025987925842309070999962783, −5.46405026093332529727973618499, −4.91857185432152356013645186868, −4.64697112936517927580279808110, −4.47880437122444323478276699536, −4.39871097506363557062342686000, −3.68071334079062017918626527098, −3.64649305687076378854128707365, −3.41330894915159412570997971449, −2.99710637502269660851331122298, −2.73156681447199638284421066344, −2.65461827261119141779263812211, −2.18290733472934049938907800144, −1.81373188029910758595425340552, −1.72287427867671709484176966283, −1.18674478056210034752297285636, −0.936863306241226625918730544336, −0.77429173659406431581889621458, 0.77429173659406431581889621458, 0.936863306241226625918730544336, 1.18674478056210034752297285636, 1.72287427867671709484176966283, 1.81373188029910758595425340552, 2.18290733472934049938907800144, 2.65461827261119141779263812211, 2.73156681447199638284421066344, 2.99710637502269660851331122298, 3.41330894915159412570997971449, 3.64649305687076378854128707365, 3.68071334079062017918626527098, 4.39871097506363557062342686000, 4.47880437122444323478276699536, 4.64697112936517927580279808110, 4.91857185432152356013645186868, 5.46405026093332529727973618499, 5.47025987925842309070999962783, 5.49076877856289621015410733327, 5.76352478425873227598209271367, 5.95935520872894079452128862703, 6.01599573298195131709137351758, 6.30223077681206188071927542378, 6.34505159661455297220667457763, 6.88769986878367372276323841186

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