Properties

Label 8-384e4-1.1-c2e4-0-1
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  + 6·9-s + 56·17-s + 4·25-s − 56·41-s − 92·49-s − 200·73-s + 27·81-s + 248·89-s − 584·97-s + 520·113-s − 388·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 336·153-s + 157-s + 163-s + 167-s + 292·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  + 2/3·9-s + 3.29·17-s + 4/25·25-s − 1.36·41-s − 1.87·49-s − 2.73·73-s + 1/3·81-s + 2.78·89-s − 6.02·97-s + 4.60·113-s − 3.20·121-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 + 2.19·153-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 1.72·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-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

\(L(\frac{3}{2})\) \(\approx\) \(2.360060021\)
\(L(\frac12)\) \(\approx\) \(2.360060021\)
\(L(2)\) 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$C_2$ \( ( 1 - p T^{2} )^{2} \)
good5$C_2^2$ \( ( 1 - 2 T^{2} + p^{4} T^{4} )^{2} \)
7$C_2^2$ \( ( 1 + 46 T^{2} + p^{4} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 + 194 T^{2} + p^{4} T^{4} )^{2} \)
13$C_2$ \( ( 1 - 22 T + p^{2} T^{2} )^{2}( 1 + 22 T + p^{2} T^{2} )^{2} \)
17$C_2$ \( ( 1 - 14 T + p^{2} T^{2} )^{4} \)
19$C_2^2$ \( ( 1 - 478 T^{2} + p^{4} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 - 482 T^{2} + p^{4} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 482 T^{2} + p^{4} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 - 1778 T^{2} + p^{4} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 - 1970 T^{2} + p^{4} T^{4} )^{2} \)
41$C_2$ \( ( 1 + 14 T + p^{2} T^{2} )^{4} \)
43$C_2^2$ \( ( 1 + 3650 T^{2} + p^{4} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 + 766 T^{2} + p^{4} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - 1730 T^{2} + p^{4} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + 4610 T^{2} + p^{4} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 - 4370 T^{2} + p^{4} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 + 866 T^{2} + p^{4} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 - 9506 T^{2} + p^{4} T^{4} )^{2} \)
73$C_2$ \( ( 1 + 50 T + p^{2} T^{2} )^{4} \)
79$C_2^2$ \( ( 1 - 12338 T^{2} + p^{4} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 + 13346 T^{2} + p^{4} T^{4} )^{2} \)
89$C_2$ \( ( 1 - 62 T + p^{2} T^{2} )^{4} \)
97$C_2$ \( ( 1 + 146 T + p^{2} T^{2} )^{4} \)
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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

−7.87300871015579292895366216876, −7.81047130018142817851123142398, −7.65877302346149446504204491785, −7.30997158540891970705700473266, −6.92490507681346613242178095855, −6.77952368631480579094103210185, −6.58149189688274469489882165739, −6.13785766961229702628623374441, −5.95234175785605594309805176434, −5.69789424295214023214015001789, −5.25072535788889615179863866723, −5.24723097026686343974978511357, −4.98472165906104631554677850699, −4.56174206951935703693893312639, −4.20552734000691682364851634956, −3.99420422629360208134696246011, −3.48388600920688756217473604560, −3.32618197500386366211747823630, −3.16920648551880259251113855575, −2.66824947096284882253026439140, −2.32246255945711719675453024730, −1.58477370440563507799928079507, −1.34286285132347209182417282391, −1.21046312263339819291986840948, −0.32390131585895781729059580845, 0.32390131585895781729059580845, 1.21046312263339819291986840948, 1.34286285132347209182417282391, 1.58477370440563507799928079507, 2.32246255945711719675453024730, 2.66824947096284882253026439140, 3.16920648551880259251113855575, 3.32618197500386366211747823630, 3.48388600920688756217473604560, 3.99420422629360208134696246011, 4.20552734000691682364851634956, 4.56174206951935703693893312639, 4.98472165906104631554677850699, 5.24723097026686343974978511357, 5.25072535788889615179863866723, 5.69789424295214023214015001789, 5.95234175785605594309805176434, 6.13785766961229702628623374441, 6.58149189688274469489882165739, 6.77952368631480579094103210185, 6.92490507681346613242178095855, 7.30997158540891970705700473266, 7.65877302346149446504204491785, 7.81047130018142817851123142398, 7.87300871015579292895366216876

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