Properties

Label 8-384e4-1.1-c4e4-0-2
Degree $8$
Conductor $21743271936$
Sign $1$
Analytic cond. $2.48257\times 10^{6}$
Root an. cond. $6.30032$
Motivic weight $4$
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  + 54·9-s − 1.35e3·17-s + 676·25-s + 2.31e3·41-s + 4.13e3·49-s + 3.49e4·73-s + 2.18e3·81-s + 3.64e3·89-s + 2.16e4·97-s − 1.26e4·113-s − 4.23e4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 7.30e4·153-s + 157-s + 163-s + 167-s + 1.06e5·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  + 2/3·9-s − 4.67·17-s + 1.08·25-s + 1.37·41-s + 1.72·49-s + 6.55·73-s + 1/3·81-s + 0.459·89-s + 2.30·97-s − 0.991·113-s − 2.89·121-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + 5.17e−5·139-s + 4.50e−5·149-s + 4.38e−5·151-s − 3.11·153-s + 4.05e−5·157-s + 3.76e−5·163-s + 3.58e−5·167-s + 3.74·169-s + 3.34e−5·173-s + 3.12e−5·179-s + 3.05e−5·181-s + 2.74e−5·191-s + 2.68e−5·193-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(2^{28} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(2.48257\times 10^{6}\)
Root analytic conductor: \(6.30032\)
Motivic weight: \(4\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{384} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{28} \cdot 3^{4} ,\ ( \ : 2, 2, 2, 2 ),\ 1 )\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(3.288169778\)
\(L(\frac12)\) \(\approx\) \(3.288169778\)
\(L(3)\) 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$C_2$ \( ( 1 - p^{3} T^{2} )^{2} \)
good5$C_2^2$ \( ( 1 - 338 T^{2} + p^{8} T^{4} )^{2} \)
7$C_2^2$ \( ( 1 - 2066 T^{2} + p^{8} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 + 21170 T^{2} + p^{8} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - 53474 T^{2} + p^{8} T^{4} )^{2} \)
17$C_2$ \( ( 1 + 338 T + p^{4} T^{2} )^{4} \)
19$C_2^2$ \( ( 1 + 260594 T^{2} + p^{8} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 - 23426 T^{2} + p^{8} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 + 271726 T^{2} + p^{8} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 - 137042 T^{2} + p^{8} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 - 3689954 T^{2} + p^{8} T^{4} )^{2} \)
41$C_2$ \( ( 1 - 578 T + p^{4} T^{2} )^{4} \)
43$C_2^2$ \( ( 1 + 2716850 T^{2} + p^{8} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 - 4933058 T^{2} + p^{8} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - 9797330 T^{2} + p^{8} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + 22798130 T^{2} + p^{8} T^{4} )^{2} \)
61$C_2$ \( ( 1 - 3794 T + p^{4} T^{2} )^{2}( 1 + 3794 T + p^{4} T^{2} )^{2} \)
67$C_2^2$ \( ( 1 - 28013710 T^{2} + p^{8} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 - 32426498 T^{2} + p^{8} T^{4} )^{2} \)
73$C_2$ \( ( 1 - 8734 T + p^{4} T^{2} )^{4} \)
79$C_2^2$ \( ( 1 + 48571438 T^{2} + p^{8} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 - 79276558 T^{2} + p^{8} T^{4} )^{2} \)
89$C_2$ \( ( 1 - 910 T + p^{4} T^{2} )^{4} \)
97$C_2$ \( ( 1 - 5422 T + p^{4} T^{2} )^{4} \)
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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

−7.58949365398757466259232689807, −7.46166024684715774792032001736, −6.77871611233654619397732711077, −6.67771946087191592682582015915, −6.58416583268060051005450190116, −6.49968589366631493045522382732, −6.37140804535537813900463707570, −5.67879865361347589743628666262, −5.49907790169617691126044761718, −5.04439148759725351036286504265, −5.00862572319347721066005396858, −4.63183664291523399896086026176, −4.40882115644136230983390299240, −4.03293527824268505358525342081, −3.86828312726056042266193906857, −3.79538142655755248563776694299, −3.04755833984427561508606737221, −2.76306106360925704755386886234, −2.34428443615831942889709409196, −2.13691012841414052899109094456, −2.09126214270688890135136915585, −1.51484997005490701423547336672, −0.817161908041554561167793361454, −0.69410615792173018095000668515, −0.29137577582836413102053743548, 0.29137577582836413102053743548, 0.69410615792173018095000668515, 0.817161908041554561167793361454, 1.51484997005490701423547336672, 2.09126214270688890135136915585, 2.13691012841414052899109094456, 2.34428443615831942889709409196, 2.76306106360925704755386886234, 3.04755833984427561508606737221, 3.79538142655755248563776694299, 3.86828312726056042266193906857, 4.03293527824268505358525342081, 4.40882115644136230983390299240, 4.63183664291523399896086026176, 5.00862572319347721066005396858, 5.04439148759725351036286504265, 5.49907790169617691126044761718, 5.67879865361347589743628666262, 6.37140804535537813900463707570, 6.49968589366631493045522382732, 6.58416583268060051005450190116, 6.67771946087191592682582015915, 6.77871611233654619397732711077, 7.46166024684715774792032001736, 7.58949365398757466259232689807

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