Properties

Label 8-12e12-1.1-c2e4-0-2
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 24·11-s + 28·25-s + 146·49-s + 72·59-s + 100·73-s + 480·83-s − 340·97-s − 600·107-s − 124·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 622·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  + 2.18·11-s + 1.11·25-s + 2.97·49-s + 1.22·59-s + 1.36·73-s + 5.78·83-s − 3.50·97-s − 5.60·107-s − 1.02·121-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 + 3.68·169-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

\(L(\frac{3}{2})\) \(\approx\) \(0.6428657327\)
\(L(\frac12)\) \(\approx\) \(0.6428657327\)
\(L(2)\) 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$C_2$ \( ( 1 - 8 T + p^{2} T^{2} )^{2}( 1 + 8 T + p^{2} T^{2} )^{2} \)
7$C_2^2$ \( ( 1 - 73 T^{2} + p^{4} T^{4} )^{2} \)
11$C_2$ \( ( 1 - 6 T + p^{2} T^{2} )^{4} \)
13$C_2^2$ \( ( 1 - 311 T^{2} + p^{4} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 - 394 T^{2} + p^{4} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 + 47 T^{2} + p^{4} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 - 86 T^{2} + p^{4} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 386 T^{2} + p^{4} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 - 1246 T^{2} + p^{4} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 - 2063 T^{2} + p^{4} T^{4} )^{2} \)
41$C_2$ \( ( 1 + p^{2} T^{2} )^{4} \)
43$C_2^2$ \( ( 1 + 3266 T^{2} + p^{4} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 - 3446 T^{2} + p^{4} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - 2018 T^{2} + p^{4} T^{4} )^{2} \)
59$C_2$ \( ( 1 - 18 T + p^{2} T^{2} )^{4} \)
61$C_2^2$ \( ( 1 - 1367 T^{2} + p^{4} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 + 2903 T^{2} + p^{4} T^{4} )^{2} \)
71$C_1$$\times$$C_1$ \( ( 1 - p T )^{4}( 1 + p T )^{4} \)
73$C_2$ \( ( 1 - 25 T + p^{2} T^{2} )^{4} \)
79$C_2^2$ \( ( 1 - 11521 T^{2} + p^{4} T^{4} )^{2} \)
83$C_2$ \( ( 1 - 120 T + p^{2} T^{2} )^{4} \)
89$C_2^2$ \( ( 1 - 8458 T^{2} + p^{4} T^{4} )^{2} \)
97$C_2$ \( ( 1 + 85 T + p^{2} T^{2} )^{4} \)
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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

−6.47563444441201173476082539906, −6.41375517829453595428098809524, −5.97150597631172319175500137879, −5.89822553355396595882968119072, −5.43023547581938228249597412206, −5.41025967108016951585593370322, −5.16808832369739981574809079486, −5.08644502037007069735321302136, −4.54614632437950421682803326819, −4.53621477639588217910757467323, −4.07572387768401907297457261914, −3.93017100228013014441800575752, −3.79032042766077811432701743923, −3.78520773864090544713815829100, −3.29491514078684583148219796061, −3.09858408023373421282963480037, −2.57351572884541179112075103749, −2.48681792633956190314374389686, −2.33064506367500722962377051374, −1.96955423347989828952921317692, −1.45828186040346089790104176557, −1.18305429127662150496365510566, −0.948337426413785936906888480537, −0.940128755938611742621601949487, −0.088058742603767786464212094786, 0.088058742603767786464212094786, 0.940128755938611742621601949487, 0.948337426413785936906888480537, 1.18305429127662150496365510566, 1.45828186040346089790104176557, 1.96955423347989828952921317692, 2.33064506367500722962377051374, 2.48681792633956190314374389686, 2.57351572884541179112075103749, 3.09858408023373421282963480037, 3.29491514078684583148219796061, 3.78520773864090544713815829100, 3.79032042766077811432701743923, 3.93017100228013014441800575752, 4.07572387768401907297457261914, 4.53621477639588217910757467323, 4.54614632437950421682803326819, 5.08644502037007069735321302136, 5.16808832369739981574809079486, 5.41025967108016951585593370322, 5.43023547581938228249597412206, 5.89822553355396595882968119072, 5.97150597631172319175500137879, 6.41375517829453595428098809524, 6.47563444441201173476082539906

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