Properties

Label 8-12e12-1.1-c1e4-0-2
Degree $8$
Conductor $8.916\times 10^{12}$
Sign $1$
Analytic cond. $36247.9$
Root an. cond. $3.71458$
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  + 14·25-s − 48·41-s − 22·49-s − 28·73-s − 48·89-s + 28·97-s + 48·113-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 52·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯
L(s)  = 1  + 14/5·25-s − 7.49·41-s − 3.14·49-s − 3.27·73-s − 5.08·89-s + 2.84·97-s + 4.51·113-s + 2.36·121-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 4·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(2^{24} \cdot 3^{12}\)
Sign: $1$
Analytic conductor: \(36247.9\)
Root analytic conductor: \(3.71458\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1728} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{24} \cdot 3^{12} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.6328007350\)
\(L(\frac12)\) \(\approx\) \(0.6328007350\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5$C_2^2$ \( ( 1 - 7 T^{2} + p^{2} T^{4} )^{2} \)
7$C_2^2$ \( ( 1 + 11 T^{2} + p^{2} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - 13 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2$ \( ( 1 - p T^{2} )^{4} \)
17$C_2$ \( ( 1 + p T^{2} )^{4} \)
19$C_2^2$ \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 + 59 T^{2} + p^{2} T^{4} )^{2} \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + 12 T + p T^{2} )^{4} \)
43$C_2^2$ \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 + 46 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - 31 T^{2} + p^{2} T^{4} )^{2} \)
59$C_2$ \( ( 1 - p T^{2} )^{4} \)
61$C_2^2$ \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 - 118 T^{2} + p^{2} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2$ \( ( 1 + 7 T + p T^{2} )^{4} \)
79$C_2^2$ \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 - 85 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2$ \( ( 1 + 12 T + p T^{2} )^{4} \)
97$C_2$ \( ( 1 - 7 T + p T^{2} )^{4} \)
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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

−6.63653783735297047534503153812, −6.58097452174808696712435518399, −6.41528448363044861096554854034, −5.89394362403349173463312635098, −5.71233569542156336132127634460, −5.69886368823295381944401824737, −5.14939937163596073075490099579, −5.13890288194213586430992207119, −4.77096149736976207352218970017, −4.75939808968392923388735647669, −4.73752149806686307151541859787, −4.20322666859465876670075240026, −4.07476870602391490463194354730, −3.46633812378234221084923337671, −3.43384910102145554252496610972, −3.22607131092012422219741672856, −3.10173292061255512976785411309, −2.84081080674783383975085386565, −2.57882297200787921592266231597, −1.88240454379502053402931426344, −1.71188981625885657743170246600, −1.65998473532987541164899010124, −1.37204922384296043560524626035, −0.72743786444423588133576311512, −0.16166907411097328322494178937, 0.16166907411097328322494178937, 0.72743786444423588133576311512, 1.37204922384296043560524626035, 1.65998473532987541164899010124, 1.71188981625885657743170246600, 1.88240454379502053402931426344, 2.57882297200787921592266231597, 2.84081080674783383975085386565, 3.10173292061255512976785411309, 3.22607131092012422219741672856, 3.43384910102145554252496610972, 3.46633812378234221084923337671, 4.07476870602391490463194354730, 4.20322666859465876670075240026, 4.73752149806686307151541859787, 4.75939808968392923388735647669, 4.77096149736976207352218970017, 5.13890288194213586430992207119, 5.14939937163596073075490099579, 5.69886368823295381944401824737, 5.71233569542156336132127634460, 5.89394362403349173463312635098, 6.41528448363044861096554854034, 6.58097452174808696712435518399, 6.63653783735297047534503153812

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