Properties

Label 8-882e4-1.1-c3e4-0-11
Degree $8$
Conductor $605165749776$
Sign $1$
Analytic cond. $7.33396\times 10^{6}$
Root an. cond. $7.21385$
Motivic weight $3$
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  + 4·2-s + 4·4-s − 16·8-s + 40·11-s − 64·16-s + 160·22-s + 96·23-s + 162·25-s + 664·29-s − 64·32-s + 156·37-s + 1.74e3·43-s + 160·44-s + 384·46-s + 648·50-s + 124·53-s + 2.65e3·58-s + 192·64-s − 1.16e3·67-s + 2.17e3·71-s + 624·74-s + 1.36e3·79-s + 6.97e3·86-s − 640·88-s + 384·92-s + 648·100-s + 496·106-s + ⋯
L(s)  = 1  + 1.41·2-s + 1/2·4-s − 0.707·8-s + 1.09·11-s − 16-s + 1.55·22-s + 0.870·23-s + 1.29·25-s + 4.25·29-s − 0.353·32-s + 0.693·37-s + 6.18·43-s + 0.548·44-s + 1.23·46-s + 1.83·50-s + 0.321·53-s + 6.01·58-s + 3/8·64-s − 2.11·67-s + 3.63·71-s + 0.980·74-s + 1.93·79-s + 8.74·86-s − 0.775·88-s + 0.435·92-s + 0.647·100-s + 0.454·106-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{8} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(7.33396\times 10^{6}\)
Root analytic conductor: \(7.21385\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{882} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{8} \cdot 7^{8} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(28.03678299\)
\(L(\frac12)\) \(\approx\) \(28.03678299\)
\(L(\frac{5}{2})\) 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$C_2$ \( ( 1 - p T + p^{2} T^{2} )^{2} \)
3 \( 1 \)
7 \( 1 \)
good5$C_2^3$ \( 1 - 162 T^{2} + 10619 T^{4} - 162 p^{6} T^{6} + p^{12} T^{8} \)
11$C_2^2$ \( ( 1 - 20 T - 931 T^{2} - 20 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 + 82 T^{2} + p^{6} T^{4} )^{2} \)
17$C_2^3$ \( 1 - 6658 T^{2} + 20191395 T^{4} - 6658 p^{6} T^{6} + p^{12} T^{8} \)
19$C_2^3$ \( 1 - 13630 T^{2} + 138731019 T^{4} - 13630 p^{6} T^{6} + p^{12} T^{8} \)
23$C_2^2$ \( ( 1 - 48 T - 9863 T^{2} - 48 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
29$C_2$ \( ( 1 - 166 T + p^{3} T^{2} )^{4} \)
31$C_2^3$ \( 1 - 16990 T^{2} - 598843581 T^{4} - 16990 p^{6} T^{6} + p^{12} T^{8} \)
37$C_2^2$ \( ( 1 - 78 T - 44569 T^{2} - 78 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 - 17390 T^{2} + p^{6} T^{4} )^{2} \)
43$C_2$ \( ( 1 - 436 T + p^{3} T^{2} )^{4} \)
47$C_2^3$ \( 1 - 165054 T^{2} + 16463607587 T^{4} - 165054 p^{6} T^{6} + p^{12} T^{8} \)
53$C_2^2$ \( ( 1 - 62 T - 145033 T^{2} - 62 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
59$C_2^3$ \( 1 + 32850 T^{2} - 41101411141 T^{4} + 32850 p^{6} T^{6} + p^{12} T^{8} \)
61$C_2^3$ \( 1 - 379954 T^{2} + 92844667755 T^{4} - 379954 p^{6} T^{6} + p^{12} T^{8} \)
67$C_2^2$ \( ( 1 + 580 T + 35637 T^{2} + 580 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
71$C_2$ \( ( 1 - 544 T + p^{3} T^{2} )^{4} \)
73$C_2^3$ \( 1 - 417586 T^{2} + 23043841107 T^{4} - 417586 p^{6} T^{6} + p^{12} T^{8} \)
79$C_2^2$ \( ( 1 - 680 T - 30639 T^{2} - 680 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 + 1104766 T^{2} + p^{6} T^{4} )^{2} \)
89$C_2^3$ \( 1 + 842862 T^{2} + 213435060083 T^{4} + 842862 p^{6} T^{6} + p^{12} T^{8} \)
97$C_2^2$ \( ( 1 + 1394146 T^{2} + p^{6} T^{4} )^{2} \)
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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

−6.79493554135649857650401334625, −6.48403248122904204447226352379, −6.31273507305632200413145822321, −6.08685302368094184914655572498, −5.96460568717282780433992665691, −5.84505398010262154759782944284, −5.42861670696303122012108369333, −5.05964019199122690336129594300, −4.88787473516630609521985663258, −4.66515685763243287263335763480, −4.62507034621758651833025703653, −4.19094983641107607890686399542, −4.15774273458987243670903734212, −3.81690643665550885140924116491, −3.53829197894345859521657427419, −3.16737201111217695423377132048, −2.94334116871317868461820499553, −2.65407077185151287912117324429, −2.51706869632508989809956534808, −2.24614074098158462520568644211, −1.75130485005546348402793536194, −0.977565322822245351735779566444, −0.960557840292143462694534675432, −0.830220883700939078431256675228, −0.55267434754114883875899763277, 0.55267434754114883875899763277, 0.830220883700939078431256675228, 0.960557840292143462694534675432, 0.977565322822245351735779566444, 1.75130485005546348402793536194, 2.24614074098158462520568644211, 2.51706869632508989809956534808, 2.65407077185151287912117324429, 2.94334116871317868461820499553, 3.16737201111217695423377132048, 3.53829197894345859521657427419, 3.81690643665550885140924116491, 4.15774273458987243670903734212, 4.19094983641107607890686399542, 4.62507034621758651833025703653, 4.66515685763243287263335763480, 4.88787473516630609521985663258, 5.05964019199122690336129594300, 5.42861670696303122012108369333, 5.84505398010262154759782944284, 5.96460568717282780433992665691, 6.08685302368094184914655572498, 6.31273507305632200413145822321, 6.48403248122904204447226352379, 6.79493554135649857650401334625

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