Properties

Label 8-12e12-1.1-c1e4-0-24
Degree $8$
Conductor $8.916\times 10^{12}$
Sign $1$
Analytic cond. $36247.9$
Root an. cond. $3.71458$
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  − 4·5-s − 6·7-s + 8·11-s + 14·13-s + 16·17-s + 6·19-s − 6·23-s − 4·25-s − 6·29-s + 8·31-s + 24·35-s + 12·37-s − 2·43-s − 2·47-s + 8·49-s + 16·53-s − 32·55-s + 12·59-s + 24·61-s − 56·65-s + 14·67-s − 48·77-s + 24·79-s − 2·83-s − 64·85-s − 84·91-s − 24·95-s + ⋯
L(s)  = 1  − 1.78·5-s − 2.26·7-s + 2.41·11-s + 3.88·13-s + 3.88·17-s + 1.37·19-s − 1.25·23-s − 4/5·25-s − 1.11·29-s + 1.43·31-s + 4.05·35-s + 1.97·37-s − 0.304·43-s − 0.291·47-s + 8/7·49-s + 2.19·53-s − 4.31·55-s + 1.56·59-s + 3.07·61-s − 6.94·65-s + 1.71·67-s − 5.47·77-s + 2.70·79-s − 0.219·83-s − 6.94·85-s − 8.80·91-s − 2.46·95-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{24} \cdot 3^{12}\)
Sign: $1$
Analytic conductor: \(36247.9\)
Root analytic conductor: \(3.71458\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1728} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{24} \cdot 3^{12} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(5.030295058\)
\(L(\frac12)\) \(\approx\) \(5.030295058\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5$D_4\times C_2$ \( 1 + 4 T + 4 p T^{2} + 52 T^{3} + 151 T^{4} + 52 p T^{5} + 4 p^{3} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
7$D_4\times C_2$ \( 1 + 6 T + 4 p T^{2} + 96 T^{3} + 291 T^{4} + 96 p T^{5} + 4 p^{3} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
11$D_4\times C_2$ \( 1 - 8 T + 41 T^{2} - 152 T^{3} + 532 T^{4} - 152 p T^{5} + 41 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2$$\times$$C_2^2$ \( ( 1 - 7 T + p T^{2} )^{2}( 1 - T^{2} + p^{2} T^{4} ) \)
17$C_2^2$ \( ( 1 - 8 T + 47 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
19$D_4\times C_2$ \( 1 - 6 T + 18 T^{2} - 132 T^{3} + 959 T^{4} - 132 p T^{5} + 18 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
23$D_4\times C_2$ \( 1 + 6 T + 52 T^{2} + 240 T^{3} + 1347 T^{4} + 240 p T^{5} + 52 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2^3$ \( 1 + 6 T + 18 T^{2} + 36 T^{3} - 457 T^{4} + 36 p T^{5} + 18 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
31$D_4\times C_2$ \( 1 - 8 T - 2 T^{2} - 32 T^{3} + 1411 T^{4} - 32 p T^{5} - 2 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
37$D_4\times C_2$ \( 1 - 12 T + 72 T^{2} - 588 T^{3} + 4658 T^{4} - 588 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2^3$ \( 1 + 73 T^{2} + 3648 T^{4} + 73 p^{2} T^{6} + p^{4} T^{8} \)
43$D_4\times C_2$ \( 1 + 2 T + 65 T^{2} + 426 T^{3} + 2744 T^{4} + 426 p T^{5} + 65 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
47$D_4\times C_2$ \( 1 + 2 T - 16 T^{2} - 148 T^{3} - 1997 T^{4} - 148 p T^{5} - 16 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
53$D_4\times C_2$ \( 1 - 16 T + 128 T^{2} - 976 T^{3} + 7378 T^{4} - 976 p T^{5} + 128 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
59$D_4\times C_2$ \( 1 - 12 T + 45 T^{2} + 828 T^{3} - 9748 T^{4} + 828 p T^{5} + 45 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
61$D_4\times C_2$ \( 1 - 24 T + 180 T^{2} + 300 T^{3} - 11209 T^{4} + 300 p T^{5} + 180 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \)
67$D_4\times C_2$ \( 1 - 14 T + 113 T^{2} - 726 T^{3} + 3848 T^{4} - 726 p T^{5} + 113 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \)
71$D_4\times C_2$ \( 1 - 156 T^{2} + 13094 T^{4} - 156 p^{2} T^{6} + p^{4} T^{8} \)
73$D_4\times C_2$ \( 1 - 158 T^{2} + 16131 T^{4} - 158 p^{2} T^{6} + p^{4} T^{8} \)
79$C_2^2$ \( ( 1 - 12 T + 65 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \)
83$C_2^3$ \( 1 + 2 T + 2 T^{2} - 328 T^{3} - 7217 T^{4} - 328 p T^{5} + 2 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2^2$ \( ( 1 - 174 T^{2} + p^{2} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 + 20 T + 109 T^{2} + 20 p T^{3} + 376 p T^{4} + 20 p^{2} T^{5} + 109 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \)
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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

−6.61566987415076685668379069099, −6.61066232324852475097844279835, −6.08645823414107913493081756111, −5.92151720725035310854128803898, −5.80621193273658517338743310043, −5.73878116097684680097118464735, −5.65372280858847588804157806025, −5.23082665244508176365377287803, −5.02388676061866932290372354591, −4.25086069460542728494530302412, −4.21977933441408093319507163962, −4.06305278238611565285353246796, −3.82016279354230835189296415412, −3.62441142702721169725508933990, −3.52233089182616022207573523700, −3.46771461088538889382176181062, −3.43193989464169651925909760574, −2.85272185069481697134385075933, −2.65949281157556167479211193929, −2.10599638341692400929609899220, −1.62724581939789917924057996070, −1.31933835114163033168861052078, −1.01724971669487656938988415798, −0.75824261685559736102049353203, −0.63382429850806899140495433891, 0.63382429850806899140495433891, 0.75824261685559736102049353203, 1.01724971669487656938988415798, 1.31933835114163033168861052078, 1.62724581939789917924057996070, 2.10599638341692400929609899220, 2.65949281157556167479211193929, 2.85272185069481697134385075933, 3.43193989464169651925909760574, 3.46771461088538889382176181062, 3.52233089182616022207573523700, 3.62441142702721169725508933990, 3.82016279354230835189296415412, 4.06305278238611565285353246796, 4.21977933441408093319507163962, 4.25086069460542728494530302412, 5.02388676061866932290372354591, 5.23082665244508176365377287803, 5.65372280858847588804157806025, 5.73878116097684680097118464735, 5.80621193273658517338743310043, 5.92151720725035310854128803898, 6.08645823414107913493081756111, 6.61066232324852475097844279835, 6.61566987415076685668379069099

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