Properties

Label 8-882e4-1.1-c2e4-0-10
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  − 4·4-s + 32·13-s + 12·16-s − 80·19-s − 16·25-s − 16·31-s + 152·37-s + 80·43-s − 128·52-s − 232·61-s − 32·64-s − 192·67-s + 96·73-s + 320·76-s + 304·79-s + 288·97-s + 64·100-s − 272·103-s − 288·109-s + 20·121-s + 64·124-s + 127-s + 131-s + 137-s + 139-s − 608·148-s + 149-s + ⋯
L(s)  = 1  − 4-s + 2.46·13-s + 3/4·16-s − 4.21·19-s − 0.639·25-s − 0.516·31-s + 4.10·37-s + 1.86·43-s − 2.46·52-s − 3.80·61-s − 1/2·64-s − 2.86·67-s + 1.31·73-s + 4.21·76-s + 3.84·79-s + 2.96·97-s + 0.639·100-s − 2.64·103-s − 2.64·109-s + 0.165·121-s + 0.516·124-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 4.10·148-s + 0.00671·149-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

\(L(\frac{3}{2})\) \(\approx\) \(1.897814591\)
\(L(\frac12)\) \(\approx\) \(1.897814591\)
\(L(2)\) 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$C_2$ \( ( 1 + p T^{2} )^{2} \)
3 \( 1 \)
7 \( 1 \)
good5$D_4\times C_2$ \( 1 + 16 T^{2} + 866 T^{4} + 16 p^{4} T^{6} + p^{8} T^{8} \)
11$D_4\times C_2$ \( 1 - 20 T^{2} + 22214 T^{4} - 20 p^{4} T^{6} + p^{8} T^{8} \)
13$D_{4}$ \( ( 1 - 16 T + 290 T^{2} - 16 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
17$D_4\times C_2$ \( 1 - 368 T^{2} + 125186 T^{4} - 368 p^{4} T^{6} + p^{8} T^{8} \)
19$C_2$ \( ( 1 + 20 T + p^{2} T^{2} )^{4} \)
23$D_4\times C_2$ \( 1 - 1652 T^{2} + 1234790 T^{4} - 1652 p^{4} T^{6} + p^{8} T^{8} \)
29$D_4\times C_2$ \( 1 - 1472 T^{2} + 1309346 T^{4} - 1472 p^{4} T^{6} + p^{8} T^{8} \)
31$D_{4}$ \( ( 1 + 8 T + 1490 T^{2} + 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
37$C_2$ \( ( 1 - 38 T + p^{2} T^{2} )^{4} \)
41$D_4\times C_2$ \( 1 - 2800 T^{2} + 4672194 T^{4} - 2800 p^{4} T^{6} + p^{8} T^{8} \)
43$D_{4}$ \( ( 1 - 40 T + 66 T^{2} - 40 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 - 4130 T^{2} + p^{4} T^{4} )^{2} \)
53$D_4\times C_2$ \( 1 + 4928 T^{2} + 21271650 T^{4} + 4928 p^{4} T^{6} + p^{8} T^{8} \)
59$D_4\times C_2$ \( 1 - 10532 T^{2} + 49098278 T^{4} - 10532 p^{4} T^{6} + p^{8} T^{8} \)
61$D_{4}$ \( ( 1 + 116 T + 9014 T^{2} + 116 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
67$D_{4}$ \( ( 1 + 96 T + 4114 T^{2} + 96 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
71$D_4\times C_2$ \( 1 - 19700 T^{2} + 147838694 T^{4} - 19700 p^{4} T^{6} + p^{8} T^{8} \)
73$D_{4}$ \( ( 1 - 48 T + 8434 T^{2} - 48 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
79$D_{4}$ \( ( 1 - 152 T + 16466 T^{2} - 152 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - 9380 T^{2} + 87552614 T^{4} - 9380 p^{4} T^{6} + p^{8} T^{8} \)
89$D_4\times C_2$ \( 1 - 12208 T^{2} + 68358210 T^{4} - 12208 p^{4} T^{6} + p^{8} T^{8} \)
97$D_{4}$ \( ( 1 - 144 T + 10450 T^{2} - 144 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
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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

−7.18894495424599368834722920403, −6.47302127890892796511219176598, −6.39581867388779198405957859918, −6.32478025212860335265152778726, −6.31090114922020771723968919292, −5.93796801670953561996555737755, −5.88854674261695690186399994311, −5.50859034353940230781116776037, −5.27579683939561764961659537325, −4.56808557060107565393695849427, −4.55910238973426438417750038752, −4.52631805254222033115541580933, −4.31819383622217424229097272028, −3.89632410405985058936055770376, −3.87090155293225851281059968014, −3.43204930669500127619164168794, −3.29645970813294176616998095564, −2.65499430239307134839998459015, −2.58594777894966329508181404296, −2.08873890762636267050547469465, −1.98163545217554938788899857921, −1.32421606524410398397416818128, −1.20517561832429915827037626936, −0.60158639314093212695543247492, −0.29130725124109142278209907130, 0.29130725124109142278209907130, 0.60158639314093212695543247492, 1.20517561832429915827037626936, 1.32421606524410398397416818128, 1.98163545217554938788899857921, 2.08873890762636267050547469465, 2.58594777894966329508181404296, 2.65499430239307134839998459015, 3.29645970813294176616998095564, 3.43204930669500127619164168794, 3.87090155293225851281059968014, 3.89632410405985058936055770376, 4.31819383622217424229097272028, 4.52631805254222033115541580933, 4.55910238973426438417750038752, 4.56808557060107565393695849427, 5.27579683939561764961659537325, 5.50859034353940230781116776037, 5.88854674261695690186399994311, 5.93796801670953561996555737755, 6.31090114922020771723968919292, 6.32478025212860335265152778726, 6.39581867388779198405957859918, 6.47302127890892796511219176598, 7.18894495424599368834722920403

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