Properties

Label 8-1536e4-1.1-c1e4-0-1
Degree $8$
Conductor $5.566\times 10^{12}$
Sign $1$
Analytic cond. $22629.4$
Root an. cond. $3.50214$
Motivic weight $1$
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  − 8·7-s − 2·9-s − 8·17-s − 16·23-s + 8·25-s + 24·31-s + 24·41-s + 16·47-s + 16·49-s + 16·63-s − 48·71-s − 16·73-s + 40·79-s + 3·81-s − 8·89-s + 8·97-s − 8·103-s + 24·113-s + 64·119-s + 36·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 16·153-s + ⋯
L(s)  = 1  − 3.02·7-s − 2/3·9-s − 1.94·17-s − 3.33·23-s + 8/5·25-s + 4.31·31-s + 3.74·41-s + 2.33·47-s + 16/7·49-s + 2.01·63-s − 5.69·71-s − 1.87·73-s + 4.50·79-s + 1/3·81-s − 0.847·89-s + 0.812·97-s − 0.788·103-s + 2.25·113-s + 5.86·119-s + 3.27·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 1.29·153-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{36} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(22629.4\)
Root analytic conductor: \(3.50214\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{36} \cdot 3^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.1578845424\)
\(L(\frac12)\) \(\approx\) \(0.1578845424\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 + T^{2} )^{2} \)
good5$D_4\times C_2$ \( 1 - 8 T^{2} + 34 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \)
7$D_{4}$ \( ( 1 + 4 T + 16 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \)
17$C_4$ \( ( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \)
23$D_{4}$ \( ( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
29$D_4\times C_2$ \( 1 - 104 T^{2} + 4354 T^{4} - 104 p^{2} T^{6} + p^{4} T^{8} \)
31$D_{4}$ \( ( 1 - 12 T + 96 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \)
37$D_4\times C_2$ \( 1 - 52 T^{2} + 1366 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \)
41$C_4$ \( ( 1 - 12 T + 86 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - 76 T^{2} + 3094 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \)
47$D_{4}$ \( ( 1 - 8 T + 38 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
53$D_4\times C_2$ \( 1 - 8 T^{2} + 4066 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \)
59$C_2^2$ \( ( 1 - 102 T^{2} + p^{2} T^{4} )^{2} \)
61$D_4\times C_2$ \( 1 - 148 T^{2} + 10870 T^{4} - 148 p^{2} T^{6} + p^{4} T^{8} \)
67$C_2^2$ \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \)
71$D_{4}$ \( ( 1 + 24 T + 278 T^{2} + 24 p T^{3} + p^{2} T^{4} )^{2} \)
73$D_{4}$ \( ( 1 + 8 T + 130 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
79$D_{4}$ \( ( 1 - 20 T + 240 T^{2} - 20 p T^{3} + p^{2} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - 68 T^{2} + 12886 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} \)
89$C_2$ \( ( 1 + 2 T + p T^{2} )^{4} \)
97$D_{4}$ \( ( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
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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

−6.62556673608918258678822105816, −6.29218109487093422444030427768, −6.28845110085632083970599126935, −6.27109582918201779910648328658, −5.99147508917696044031915295733, −5.81349288483746806590914092557, −5.75635496494063551230845869641, −5.28032299262245179452380244487, −4.78458463444902388236506512588, −4.69529917817382954550812740375, −4.38208042267519582795814839890, −4.26759341834487760253298501176, −4.20922803543672862546530609399, −3.85982861771932665568151632930, −3.39775947061958694993848936619, −3.26220628638032236354162177701, −3.02867836740251332675597581298, −2.64097456115092330237852405054, −2.59062824636754290046267531710, −2.46685874797301193814356934110, −2.09885868624070547911338808595, −1.57376168064719150215121700835, −0.919227192973046285345743900194, −0.74591441958885828770525285792, −0.10594265251544755500363837976, 0.10594265251544755500363837976, 0.74591441958885828770525285792, 0.919227192973046285345743900194, 1.57376168064719150215121700835, 2.09885868624070547911338808595, 2.46685874797301193814356934110, 2.59062824636754290046267531710, 2.64097456115092330237852405054, 3.02867836740251332675597581298, 3.26220628638032236354162177701, 3.39775947061958694993848936619, 3.85982861771932665568151632930, 4.20922803543672862546530609399, 4.26759341834487760253298501176, 4.38208042267519582795814839890, 4.69529917817382954550812740375, 4.78458463444902388236506512588, 5.28032299262245179452380244487, 5.75635496494063551230845869641, 5.81349288483746806590914092557, 5.99147508917696044031915295733, 6.27109582918201779910648328658, 6.28845110085632083970599126935, 6.29218109487093422444030427768, 6.62556673608918258678822105816

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