Properties

Label 8-384e4-1.1-c5e4-0-0
Degree $8$
Conductor $21743271936$
Sign $1$
Analytic cond. $1.43868\times 10^{7}$
Root an. cond. $7.84776$
Motivic weight $5$
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  − 32·7-s − 162·9-s + 1.84e3·17-s − 7.16e3·23-s + 9.74e3·25-s − 7.07e3·31-s + 5.03e3·41-s − 9.15e3·47-s − 1.77e4·49-s + 5.18e3·63-s − 1.51e5·71-s + 3.42e4·73-s − 2.00e5·79-s + 1.96e4·81-s − 1.80e5·89-s − 1.99e5·97-s − 6.79e5·103-s − 4.98e4·113-s − 5.91e4·119-s + 5.53e5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 2.99e5·153-s + ⋯
L(s)  = 1  − 0.246·7-s − 2/3·9-s + 1.55·17-s − 2.82·23-s + 3.11·25-s − 1.32·31-s + 0.467·41-s − 0.604·47-s − 1.05·49-s + 0.164·63-s − 3.55·71-s + 0.751·73-s − 3.62·79-s + 1/3·81-s − 2.41·89-s − 2.15·97-s − 6.30·103-s − 0.367·113-s − 0.382·119-s + 3.43·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s − 1.03·153-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{28} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(1.43868\times 10^{7}\)
Root analytic conductor: \(7.84776\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{384} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{28} \cdot 3^{4} ,\ ( \ : 5/2, 5/2, 5/2, 5/2 ),\ 1 )\)

Particular Values

\(L(3)\) \(\approx\) \(0.2669631136\)
\(L(\frac12)\) \(\approx\) \(0.2669631136\)
\(L(\frac{7}{2})\) 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$C_2$ \( ( 1 + p^{4} T^{2} )^{2} \)
good5$D_4\times C_2$ \( 1 - 9748 T^{2} + 41725526 T^{4} - 9748 p^{10} T^{6} + p^{20} T^{8} \)
7$D_{4}$ \( ( 1 + 16 T + 9278 T^{2} + 16 p^{5} T^{3} + p^{10} T^{4} )^{2} \)
11$D_4\times C_2$ \( 1 - 553804 T^{2} + 126701631830 T^{4} - 553804 p^{10} T^{6} + p^{20} T^{8} \)
13$D_4\times C_2$ \( 1 - 1079252 T^{2} + 557988097398 T^{4} - 1079252 p^{10} T^{6} + p^{20} T^{8} \)
17$D_{4}$ \( ( 1 - 924 T + 414054 T^{2} - 924 p^{5} T^{3} + p^{10} T^{4} )^{2} \)
19$D_4\times C_2$ \( 1 - 8230380 T^{2} + 28977086949238 T^{4} - 8230380 p^{10} T^{6} + p^{20} T^{8} \)
23$D_{4}$ \( ( 1 + 3584 T + 15611566 T^{2} + 3584 p^{5} T^{3} + p^{10} T^{4} )^{2} \)
29$D_4\times C_2$ \( 1 - 36903156 T^{2} + 892166002897910 T^{4} - 36903156 p^{10} T^{6} + p^{20} T^{8} \)
31$D_{4}$ \( ( 1 + 3536 T + 38130350 T^{2} + 3536 p^{5} T^{3} + p^{10} T^{4} )^{2} \)
37$D_4\times C_2$ \( 1 - 220032564 T^{2} + 21466526966405398 T^{4} - 220032564 p^{10} T^{6} + p^{20} T^{8} \)
41$D_{4}$ \( ( 1 - 2516 T + 67624822 T^{2} - 2516 p^{5} T^{3} + p^{10} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - 77492172 T^{2} + 41298446982584278 T^{4} - 77492172 p^{10} T^{6} + p^{20} T^{8} \)
47$D_{4}$ \( ( 1 + 4576 T + 377252254 T^{2} + 4576 p^{5} T^{3} + p^{10} T^{4} )^{2} \)
53$D_4\times C_2$ \( 1 - 1670187732 T^{2} + 1047155709832775318 T^{4} - 1670187732 p^{10} T^{6} + p^{20} T^{8} \)
59$D_4\times C_2$ \( 1 - 1321495820 T^{2} + 1013409615111355158 T^{4} - 1321495820 p^{10} T^{6} + p^{20} T^{8} \)
61$D_4\times C_2$ \( 1 - 2092476436 T^{2} + 2164810071205318326 T^{4} - 2092476436 p^{10} T^{6} + p^{20} T^{8} \)
67$D_4\times C_2$ \( 1 - 2098688428 T^{2} + 4067732954700328758 T^{4} - 2098688428 p^{10} T^{6} + p^{20} T^{8} \)
71$D_{4}$ \( ( 1 + 75520 T + 4788320398 T^{2} + 75520 p^{5} T^{3} + p^{10} T^{4} )^{2} \)
73$D_{4}$ \( ( 1 - 17100 T + 1464583286 T^{2} - 17100 p^{5} T^{3} + p^{10} T^{4} )^{2} \)
79$D_{4}$ \( ( 1 + 100496 T + 8128509326 T^{2} + 100496 p^{5} T^{3} + p^{10} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - 3511348972 T^{2} + 9858377737524874358 T^{4} - 3511348972 p^{10} T^{6} + p^{20} T^{8} \)
89$D_{4}$ \( ( 1 + 90356 T + 13201612438 T^{2} + 90356 p^{5} T^{3} + p^{10} T^{4} )^{2} \)
97$D_{4}$ \( ( 1 + 99908 T + 11886506054 T^{2} + 99908 p^{5} T^{3} + p^{10} T^{4} )^{2} \)
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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

−7.24653614201376028337628098516, −7.12927109696533975816052880496, −7.09496094244350952175734909167, −6.45764845212004375145461191388, −6.39902179272026848982300526146, −6.02742535107219663986394098750, −5.80513324535744568390869668175, −5.65548632418545377109700889167, −5.26476748203637357605163100306, −5.19364906402041831822506726327, −4.80449811937312266101653520386, −4.37003113080977011660631418273, −4.15087968016217228776797774586, −3.88328040356417913480644467850, −3.74761369042458105810201377347, −3.01322387636162556858648444887, −2.97991706473567891826483484906, −2.75559932954945987183461356030, −2.65662159063510492265622366879, −1.77024513102685697523379235064, −1.62373907443359601486798040574, −1.39738305990691611040881723734, −1.05360828453635942957651328033, −0.42262216125422588966083227915, −0.084169289427573738078334534196, 0.084169289427573738078334534196, 0.42262216125422588966083227915, 1.05360828453635942957651328033, 1.39738305990691611040881723734, 1.62373907443359601486798040574, 1.77024513102685697523379235064, 2.65662159063510492265622366879, 2.75559932954945987183461356030, 2.97991706473567891826483484906, 3.01322387636162556858648444887, 3.74761369042458105810201377347, 3.88328040356417913480644467850, 4.15087968016217228776797774586, 4.37003113080977011660631418273, 4.80449811937312266101653520386, 5.19364906402041831822506726327, 5.26476748203637357605163100306, 5.65548632418545377109700889167, 5.80513324535744568390869668175, 6.02742535107219663986394098750, 6.39902179272026848982300526146, 6.45764845212004375145461191388, 7.09496094244350952175734909167, 7.12927109696533975816052880496, 7.24653614201376028337628098516

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