Properties

Label 8-12e12-1.1-c2e4-0-14
Degree $8$
Conductor $8.916\times 10^{12}$
Sign $1$
Analytic cond. $4.91490\times 10^{6}$
Root an. cond. $6.86182$
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  + 24·17-s + 76·25-s − 192·41-s + 194·49-s + 196·73-s + 264·89-s + 428·97-s + 600·113-s + 116·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 670·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  + 1.41·17-s + 3.03·25-s − 4.68·41-s + 3.95·49-s + 2.68·73-s + 2.96·89-s + 4.41·97-s + 5.30·113-s + 0.958·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 3.96·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{24} \cdot 3^{12}\)
Sign: $1$
Analytic conductor: \(4.91490\times 10^{6}\)
Root analytic conductor: \(6.86182\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1728} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{24} \cdot 3^{12} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(6.139348412\)
\(L(\frac12)\) \(\approx\) \(6.139348412\)
\(L(2)\) 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 \( 1 \)
3 \( 1 \)
good5$C_2^2$ \( ( 1 - 38 T^{2} + p^{4} T^{4} )^{2} \)
7$C_2^2$ \( ( 1 - 97 T^{2} + p^{4} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - 58 T^{2} + p^{4} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - 335 T^{2} + p^{4} T^{4} )^{2} \)
17$C_2$ \( ( 1 - 6 T + p^{2} T^{2} )^{4} \)
19$C_2^2$ \( ( 1 + 719 T^{2} + p^{4} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 - 158 T^{2} + p^{4} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 1250 T^{2} + p^{4} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 - 1726 T^{2} + p^{4} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 - 2375 T^{2} + p^{4} T^{4} )^{2} \)
41$C_2$ \( ( 1 + 48 T + p^{2} T^{2} )^{4} \)
43$C_2^2$ \( ( 1 + 3266 T^{2} + p^{4} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 - 62 T^{2} + p^{4} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - 3266 T^{2} + p^{4} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + 5990 T^{2} + p^{4} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 - 5567 T^{2} + p^{4} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 + 5303 T^{2} + p^{4} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 - 7778 T^{2} + p^{4} T^{4} )^{2} \)
73$C_2$ \( ( 1 - 49 T + p^{2} T^{2} )^{4} \)
79$C_2^2$ \( ( 1 - 5593 T^{2} + p^{4} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 + 13586 T^{2} + p^{4} T^{4} )^{2} \)
89$C_2$ \( ( 1 - 66 T + p^{2} T^{2} )^{4} \)
97$C_2$ \( ( 1 - 107 T + p^{2} T^{2} )^{4} \)
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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

−6.61225326912235546332118603996, −6.24561322722313000841246326000, −5.97935966202876790794441695732, −5.83316525517568769476645591477, −5.62044634308514273334210133300, −5.07317408768245201954419326770, −5.04398864026024653551730995206, −5.04155306834375656935811507429, −4.98905928838128514454043273786, −4.60916149239987133887358081849, −4.06417002691866182933559828920, −4.05991413309641783541491976341, −3.75621958791742054510380953727, −3.30065820220666940034144252849, −3.24209689520940566963425092489, −3.23114777176107155036790157718, −2.94364649181144867130936339030, −2.35069469468591793918603266635, −2.16284912108941812069817304217, −1.98175002532849832403377901390, −1.74670206546397509577829715902, −1.00274658529169511757668135364, −0.908430742446098265536796631963, −0.864995184155536630805374949012, −0.31830479102582335609869801709, 0.31830479102582335609869801709, 0.864995184155536630805374949012, 0.908430742446098265536796631963, 1.00274658529169511757668135364, 1.74670206546397509577829715902, 1.98175002532849832403377901390, 2.16284912108941812069817304217, 2.35069469468591793918603266635, 2.94364649181144867130936339030, 3.23114777176107155036790157718, 3.24209689520940566963425092489, 3.30065820220666940034144252849, 3.75621958791742054510380953727, 4.05991413309641783541491976341, 4.06417002691866182933559828920, 4.60916149239987133887358081849, 4.98905928838128514454043273786, 5.04155306834375656935811507429, 5.04398864026024653551730995206, 5.07317408768245201954419326770, 5.62044634308514273334210133300, 5.83316525517568769476645591477, 5.97935966202876790794441695732, 6.24561322722313000841246326000, 6.61225326912235546332118603996

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