Properties

Label 8-882e4-1.1-c2e4-0-14
Degree $8$
Conductor $605165749776$
Sign $1$
Analytic cond. $333591.$
Root an. cond. $4.90232$
Motivic weight $2$
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  − 2·4-s − 102·19-s − 26·25-s + 42·31-s + 94·37-s + 124·43-s + 288·61-s + 8·64-s + 62·67-s + 282·73-s + 204·76-s − 82·79-s + 52·100-s + 102·103-s − 338·109-s − 46·121-s − 84·124-s + 127-s + 131-s + 137-s + 139-s − 188·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  − 1/2·4-s − 5.36·19-s − 1.03·25-s + 1.35·31-s + 2.54·37-s + 2.88·43-s + 4.72·61-s + 1/8·64-s + 0.925·67-s + 3.86·73-s + 2.68·76-s − 1.03·79-s + 0.519·100-s + 0.990·103-s − 3.10·109-s − 0.380·121-s − 0.677·124-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 1.27·148-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{8} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(333591.\)
Root analytic conductor: \(4.90232\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{882} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{8} \cdot 7^{8} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.563257180\)
\(L(\frac12)\) \(\approx\) \(3.563257180\)
\(L(2)\) 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$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
3 \( 1 \)
7 \( 1 \)
good5$C_2^3$ \( 1 + 26 T^{2} + 51 T^{4} + 26 p^{4} T^{6} + p^{8} T^{8} \)
11$C_2^3$ \( 1 + 46 T^{2} - 12525 T^{4} + 46 p^{4} T^{6} + p^{8} T^{8} \)
13$C_2^2$ \( ( 1 - 335 T^{2} + p^{4} T^{4} )^{2} \)
17$C_2^3$ \( 1 + 554 T^{2} + 223395 T^{4} + 554 p^{4} T^{6} + p^{8} T^{8} \)
19$C_2^2$ \( ( 1 + 51 T + 1228 T^{2} + 51 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
23$C_2^3$ \( 1 - 986 T^{2} + 692355 T^{4} - 986 p^{4} T^{6} + p^{8} T^{8} \)
29$C_2^2$ \( ( 1 + 530 T^{2} + p^{4} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 - 21 T + 1108 T^{2} - 21 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
37$C_2$ \( ( 1 - 73 T + p^{2} T^{2} )^{2}( 1 + 26 T + p^{2} T^{2} )^{2} \)
41$C_2^2$ \( ( 1 + 1342 T^{2} + p^{4} T^{4} )^{2} \)
43$C_2$ \( ( 1 - 31 T + p^{2} T^{2} )^{4} \)
47$C_2^3$ \( 1 - 2518 T^{2} + 1460643 T^{4} - 2518 p^{4} T^{6} + p^{8} T^{8} \)
53$C_2^3$ \( 1 + 214 T^{2} - 7844685 T^{4} + 214 p^{4} T^{6} + p^{8} T^{8} \)
59$C_2^3$ \( 1 + 26 T^{2} - 12116685 T^{4} + 26 p^{4} T^{6} + p^{8} T^{8} \)
61$C_2^2$ \( ( 1 - 144 T + 10633 T^{2} - 144 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 - 31 T - 3528 T^{2} - 31 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 + 6554 T^{2} + p^{4} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 - 141 T + 11956 T^{2} - 141 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
79$C_2^2$ \( ( 1 + 41 T - 4560 T^{2} + 41 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 - 13754 T^{2} + p^{4} T^{4} )^{2} \)
89$C_2^3$ \( 1 + 12386 T^{2} + 90670755 T^{4} + 12386 p^{4} T^{6} + p^{8} T^{8} \)
97$C_2^2$ \( ( 1 - 17090 T^{2} + p^{4} T^{4} )^{2} \)
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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

−6.80656008296706927879620323803, −6.72290373123749639508824393835, −6.61449082392830991980332719153, −6.52437680576488886989694663474, −6.06783796523835425224050753694, −6.01543365312909813469146378450, −5.75625027341597001812777971394, −5.49518987189293953616067051182, −5.22570592076553462478790842951, −4.84410008729259958665389450979, −4.49044856470717826761207405803, −4.47801955714005137688225392233, −4.12344294046862948424581976053, −4.03516007467410770028037376493, −3.83101121966491864201313134781, −3.69195176486464609392897311193, −2.86332636835051522706833611320, −2.80227115973976438946350686851, −2.27731864766174494416540554820, −2.21455063250295744442254762043, −2.13940729823331137965930335936, −1.66866018634715994168469141536, −0.73537762718519905328741362488, −0.70451896212788226893233107475, −0.44667277464964564132988662638, 0.44667277464964564132988662638, 0.70451896212788226893233107475, 0.73537762718519905328741362488, 1.66866018634715994168469141536, 2.13940729823331137965930335936, 2.21455063250295744442254762043, 2.27731864766174494416540554820, 2.80227115973976438946350686851, 2.86332636835051522706833611320, 3.69195176486464609392897311193, 3.83101121966491864201313134781, 4.03516007467410770028037376493, 4.12344294046862948424581976053, 4.47801955714005137688225392233, 4.49044856470717826761207405803, 4.84410008729259958665389450979, 5.22570592076553462478790842951, 5.49518987189293953616067051182, 5.75625027341597001812777971394, 6.01543365312909813469146378450, 6.06783796523835425224050753694, 6.52437680576488886989694663474, 6.61449082392830991980332719153, 6.72290373123749639508824393835, 6.80656008296706927879620323803

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