Properties

Degree $8$
Conductor $41006250000$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 5·5-s − 12·7-s + 5·10-s − 3·11-s − 13-s + 12·14-s − 3·17-s − 10·19-s + 3·22-s − 9·23-s + 10·25-s + 26-s − 15·29-s + 3·31-s + 32-s + 3·34-s + 60·35-s − 17·37-s + 10·38-s − 13·41-s − 16·43-s + 9·46-s − 23·47-s + 62·49-s − 10·50-s + 16·53-s + ⋯
L(s)  = 1  − 0.707·2-s − 2.23·5-s − 4.53·7-s + 1.58·10-s − 0.904·11-s − 0.277·13-s + 3.20·14-s − 0.727·17-s − 2.29·19-s + 0.639·22-s − 1.87·23-s + 2·25-s + 0.196·26-s − 2.78·29-s + 0.538·31-s + 0.176·32-s + 0.514·34-s + 10.1·35-s − 2.79·37-s + 1.62·38-s − 2.03·41-s − 2.43·43-s + 1.32·46-s − 3.35·47-s + 62/7·49-s − 1.41·50-s + 2.19·53-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{8} \cdot 5^{8}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{450} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{4} \cdot 3^{8} \cdot 5^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

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$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
3 \( 1 \)
5$C_4$ \( 1 + p T + 3 p T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
good7$C_2$ \( ( 1 + 3 T + p T^{2} )^{4} \)
11$C_2^2:C_4$ \( 1 + 3 T + 8 T^{2} + 51 T^{3} + 265 T^{4} + 51 p T^{5} + 8 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
13$C_4\times C_2$ \( 1 + T - 12 T^{2} - 25 T^{3} + 131 T^{4} - 25 p T^{5} - 12 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
17$C_2^2:C_4$ \( 1 + 3 T + 37 T^{2} + 45 T^{3} + 676 T^{4} + 45 p T^{5} + 37 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2^2:C_4$ \( 1 + 10 T + 21 T^{2} - 70 T^{3} - 469 T^{4} - 70 p T^{5} + 21 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2^2:C_4$ \( 1 + 9 T + 8 T^{2} - 195 T^{3} - 1259 T^{4} - 195 p T^{5} + 8 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2^2:C_4$ \( 1 + 15 T + 56 T^{2} - 315 T^{3} - 3629 T^{4} - 315 p T^{5} + 56 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2^2:C_4$ \( 1 - 3 T - 12 T^{2} - 131 T^{3} + 1365 T^{4} - 131 p T^{5} - 12 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2^2:C_4$ \( 1 + 17 T + 147 T^{2} + 1135 T^{3} + 7976 T^{4} + 1135 p T^{5} + 147 p^{2} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2^2:C_4$ \( 1 + 13 T + 28 T^{2} - 169 T^{3} - 945 T^{4} - 169 p T^{5} + 28 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} \)
43$D_{4}$ \( ( 1 + 8 T + 97 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
47$C_2^2:C_4$ \( 1 + 23 T + 202 T^{2} + 925 T^{3} + 4101 T^{4} + 925 p T^{5} + 202 p^{2} T^{6} + 23 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2^2:C_4$ \( 1 - 16 T + 43 T^{2} + 700 T^{3} - 7959 T^{4} + 700 p T^{5} + 43 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
59$C_4\times C_2$ \( 1 - 10 T + T^{2} - 200 T^{3} + 5061 T^{4} - 200 p T^{5} + p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2^2:C_4$ \( 1 + 2 T + 3 T^{2} + 424 T^{3} + 4265 T^{4} + 424 p T^{5} + 3 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2^2:C_4$ \( 1 - 13 T + 12 T^{2} + 895 T^{3} - 9919 T^{4} + 895 p T^{5} + 12 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \)
71$C_4\times C_2$ \( 1 + 3 T - 62 T^{2} - 399 T^{3} + 3205 T^{4} - 399 p T^{5} - 62 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2^2:C_4$ \( 1 - 14 T + 63 T^{2} - 850 T^{3} + 12521 T^{4} - 850 p T^{5} + 63 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2^2:C_4$ \( 1 - 10 T - 39 T^{2} + 10 p T^{3} - 2839 T^{4} + 10 p^{2} T^{5} - 39 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2^2:C_4$ \( 1 - T + 58 T^{2} - 335 T^{3} + 7041 T^{4} - 335 p T^{5} + 58 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
89$C_4\times C_2$ \( 1 + 10 T - 29 T^{2} + 200 T^{3} + 10101 T^{4} + 200 p T^{5} - 29 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2^2:C_4$ \( 1 + 22 T + 207 T^{2} + 2420 T^{3} + 31541 T^{4} + 2420 p T^{5} + 207 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8} \)
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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

−8.609467959933360660663665116311, −8.230378387629925224543962587662, −8.014113829548500310170152011149, −7.82348501493584776535769314511, −7.80603466757305555041438715132, −7.00440801349771244598839350277, −6.88384144510362469895364221631, −6.83841495490255345707359449930, −6.78583686158399291084737199917, −6.42718425893663748905887828693, −6.31584176006740511372226371408, −5.92830136798524843262160322659, −5.52780472724710961198477659242, −5.17859677913348361385269910383, −5.14148103799231013348664421939, −4.41762481619095627401582284407, −4.34500215480399794681566777645, −3.67781629527325832773666766864, −3.58858543393454466736065080956, −3.58751192218407239680299447081, −3.48093925008119709084154433439, −3.05000763854884951145394741735, −2.49689770459919310804008872713, −2.07865427878540105637347397105, −1.86160265433087180604029033674, 0, 0, 0, 0, 1.86160265433087180604029033674, 2.07865427878540105637347397105, 2.49689770459919310804008872713, 3.05000763854884951145394741735, 3.48093925008119709084154433439, 3.58751192218407239680299447081, 3.58858543393454466736065080956, 3.67781629527325832773666766864, 4.34500215480399794681566777645, 4.41762481619095627401582284407, 5.14148103799231013348664421939, 5.17859677913348361385269910383, 5.52780472724710961198477659242, 5.92830136798524843262160322659, 6.31584176006740511372226371408, 6.42718425893663748905887828693, 6.78583686158399291084737199917, 6.83841495490255345707359449930, 6.88384144510362469895364221631, 7.00440801349771244598839350277, 7.80603466757305555041438715132, 7.82348501493584776535769314511, 8.014113829548500310170152011149, 8.230378387629925224543962587662, 8.609467959933360660663665116311

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