Properties

Label 8-384e4-1.1-c2e4-0-3
Degree $8$
Conductor $21743271936$
Sign $1$
Analytic cond. $11985.7$
Root an. cond. $3.23469$
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  + 16·5-s + 2·9-s + 60·25-s + 208·29-s + 32·45-s − 36·49-s + 80·53-s − 200·73-s − 77·81-s + 200·97-s − 368·101-s − 444·121-s − 720·125-s + 127-s + 131-s + 137-s + 139-s + 3.32e3·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 36·169-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  + 16/5·5-s + 2/9·9-s + 12/5·25-s + 7.17·29-s + 0.711·45-s − 0.734·49-s + 1.50·53-s − 2.73·73-s − 0.950·81-s + 2.06·97-s − 3.64·101-s − 3.66·121-s − 5.75·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 22.9·145-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 0.213·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{28} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(11985.7\)
Root analytic conductor: \(3.23469\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{384} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{28} \cdot 3^{4} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(8.756648262\)
\(L(\frac12)\) \(\approx\) \(8.756648262\)
\(L(2)\) 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$C_2^2$ \( 1 - 2 T^{2} + p^{4} T^{4} \)
good5$C_2$ \( ( 1 - 4 T + p^{2} T^{2} )^{4} \)
7$C_2^2$ \( ( 1 + 18 T^{2} + p^{4} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 + 222 T^{2} + p^{4} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - 18 T^{2} + p^{4} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 - 258 T^{2} + p^{4} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 - 322 T^{2} + p^{4} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 - 802 T^{2} + p^{4} T^{4} )^{2} \)
29$C_2$ \( ( 1 - 52 T + p^{2} T^{2} )^{4} \)
31$C_2^2$ \( ( 1 + 1202 T^{2} + p^{4} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 + 142 T^{2} + p^{4} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 - 2082 T^{2} + p^{4} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 - 2402 T^{2} + p^{4} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 - 322 T^{2} + p^{4} T^{4} )^{2} \)
53$C_2$ \( ( 1 - 20 T + p^{2} T^{2} )^{4} \)
59$C_2^2$ \( ( 1 - 3618 T^{2} + p^{4} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 - 7122 T^{2} + p^{4} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 - 7042 T^{2} + p^{4} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 - 3682 T^{2} + p^{4} T^{4} )^{2} \)
73$C_2$ \( ( 1 + 50 T + p^{2} T^{2} )^{4} \)
79$C_2^2$ \( ( 1 + 6002 T^{2} + p^{4} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 + 3198 T^{2} + p^{4} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 + 10078 T^{2} + p^{4} T^{4} )^{2} \)
97$C_2$ \( ( 1 - 50 T + p^{2} T^{2} )^{4} \)
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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

−8.045928238554707265765580298313, −7.69914222225944950952510561439, −7.62384514662258914622161248502, −7.08104155450224709527964521511, −6.78080163761335159740326129856, −6.53362004306384823118236054964, −6.48310691140718284670170654128, −6.16022209912733128996485862402, −6.11202549994904674914148791954, −5.75575933106290333596706400522, −5.34167209149257630168974521349, −5.31230260011318237206989996188, −4.93895365919870181489582044968, −4.60210958180190362129053069945, −4.42830256789626795622508574274, −3.97869204400238422174435739666, −3.76308964778912732291966214090, −2.81650044889059464762199055879, −2.78701051043712922725451476046, −2.76146658519514753703866090751, −2.34673273055620304021851846479, −1.78058592660570432091408805083, −1.33722842425845759801523838001, −1.32887450225003979529010354762, −0.57164799000453321654144811090, 0.57164799000453321654144811090, 1.32887450225003979529010354762, 1.33722842425845759801523838001, 1.78058592660570432091408805083, 2.34673273055620304021851846479, 2.76146658519514753703866090751, 2.78701051043712922725451476046, 2.81650044889059464762199055879, 3.76308964778912732291966214090, 3.97869204400238422174435739666, 4.42830256789626795622508574274, 4.60210958180190362129053069945, 4.93895365919870181489582044968, 5.31230260011318237206989996188, 5.34167209149257630168974521349, 5.75575933106290333596706400522, 6.11202549994904674914148791954, 6.16022209912733128996485862402, 6.48310691140718284670170654128, 6.53362004306384823118236054964, 6.78080163761335159740326129856, 7.08104155450224709527964521511, 7.62384514662258914622161248502, 7.69914222225944950952510561439, 8.045928238554707265765580298313

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