Properties

Label 8-320e4-1.1-c7e4-0-8
Degree $8$
Conductor $10485760000$
Sign $1$
Analytic cond. $9.98529\times 10^{7}$
Root an. cond. $9.99816$
Motivic weight $7$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 500·5-s − 3.35e3·9-s − 7.51e3·13-s + 2.80e4·17-s + 1.56e5·25-s + 5.79e4·29-s + 3.49e4·37-s + 6.54e5·41-s + 1.67e6·45-s − 6.67e5·49-s − 3.51e6·53-s − 8.75e6·61-s + 3.75e6·65-s + 4.28e6·73-s + 5.85e6·81-s − 1.40e7·85-s + 5.50e6·89-s + 2.15e7·97-s − 2.66e7·101-s − 6.64e7·109-s + 3.18e6·113-s + 2.52e7·117-s − 5.27e7·121-s − 3.90e7·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  − 1.78·5-s − 1.53·9-s − 0.948·13-s + 1.38·17-s + 2·25-s + 0.441·29-s + 0.113·37-s + 1.48·41-s + 2.74·45-s − 0.811·49-s − 3.24·53-s − 4.93·61-s + 1.69·65-s + 1.28·73-s + 1.22·81-s − 2.47·85-s + 0.827·89-s + 2.39·97-s − 2.57·101-s − 4.91·109-s + 0.207·113-s + 1.45·117-s − 2.70·121-s − 1.78·125-s − 0.789·145-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{24} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(9.98529\times 10^{7}\)
Root analytic conductor: \(9.99816\)
Motivic weight: \(7\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{24} \cdot 5^{4} ,\ ( \ : 7/2, 7/2, 7/2, 7/2 ),\ 1 )\)

Particular Values

\(L(4)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{9}{2})\) 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 \( 1 \)
5$C_1$ \( ( 1 + p^{3} T )^{4} \)
good3$C_2^2 \wr C_2$ \( 1 + 3356 T^{2} + 600454 p^{2} T^{4} + 3356 p^{14} T^{6} + p^{28} T^{8} \)
7$C_2^2 \wr C_2$ \( 1 + 667980 T^{2} - 66790330298 T^{4} + 667980 p^{14} T^{6} + p^{28} T^{8} \)
11$C_2^2 \wr C_2$ \( 1 + 52768972 T^{2} + 1366013927988662 T^{4} + 52768972 p^{14} T^{6} + p^{28} T^{8} \)
13$D_{4}$ \( ( 1 + 3756 T + 122046382 T^{2} + 3756 p^{7} T^{3} + p^{14} T^{4} )^{2} \)
17$D_{4}$ \( ( 1 - 14020 T + 25535590 T^{2} - 14020 p^{7} T^{3} + p^{14} T^{4} )^{2} \)
19$C_2^2 \wr C_2$ \( 1 + 3093748716 T^{2} + 3936208542354106006 T^{4} + 3093748716 p^{14} T^{6} + p^{28} T^{8} \)
23$C_2^2 \wr C_2$ \( 1 + 1440463116 T^{2} + 21742722190819001606 T^{4} + 1440463116 p^{14} T^{6} + p^{28} T^{8} \)
29$D_{4}$ \( ( 1 - 28996 T + 31332817198 T^{2} - 28996 p^{7} T^{3} + p^{14} T^{4} )^{2} \)
31$C_2^2 \wr C_2$ \( 1 + 56799681852 T^{2} + \)\(22\!\cdots\!82\)\( T^{4} + 56799681852 p^{14} T^{6} + p^{28} T^{8} \)
37$D_{4}$ \( ( 1 - 17476 T + 49245071006 T^{2} - 17476 p^{7} T^{3} + p^{14} T^{4} )^{2} \)
41$D_{4}$ \( ( 1 - 327012 T + 212212629622 T^{2} - 327012 p^{7} T^{3} + p^{14} T^{4} )^{2} \)
43$C_2^2 \wr C_2$ \( 1 + 957482635260 T^{2} + \)\(37\!\cdots\!42\)\( T^{4} + 957482635260 p^{14} T^{6} + p^{28} T^{8} \)
47$C_2^2 \wr C_2$ \( 1 + 504600011180 T^{2} + \)\(88\!\cdots\!22\)\( T^{4} + 504600011180 p^{14} T^{6} + p^{28} T^{8} \)
53$D_{4}$ \( ( 1 + 1757596 T + 2828477254078 T^{2} + 1757596 p^{7} T^{3} + p^{14} T^{4} )^{2} \)
59$C_2^2 \wr C_2$ \( 1 - 4962689795252 T^{2} + \)\(18\!\cdots\!82\)\( T^{4} - 4962689795252 p^{14} T^{6} + p^{28} T^{8} \)
61$D_{4}$ \( ( 1 + 4376428 T + 9698242644942 T^{2} + 4376428 p^{7} T^{3} + p^{14} T^{4} )^{2} \)
67$C_2^2 \wr C_2$ \( 1 + 353773151260 T^{2} + \)\(64\!\cdots\!82\)\( T^{4} + 353773151260 p^{14} T^{6} + p^{28} T^{8} \)
71$C_2^2 \wr C_2$ \( 1 + 444390087812 p T^{2} + \)\(41\!\cdots\!22\)\( T^{4} + 444390087812 p^{15} T^{6} + p^{28} T^{8} \)
73$D_{4}$ \( ( 1 - 2142420 T + 22759658714710 T^{2} - 2142420 p^{7} T^{3} + p^{14} T^{4} )^{2} \)
79$C_2^2 \wr C_2$ \( 1 + 72988846160828 T^{2} + \)\(20\!\cdots\!62\)\( T^{4} + 72988846160828 p^{14} T^{6} + p^{28} T^{8} \)
83$C_2^2 \wr C_2$ \( 1 - 47616762572324 T^{2} + \)\(17\!\cdots\!46\)\( T^{4} - 47616762572324 p^{14} T^{6} + p^{28} T^{8} \)
89$D_{4}$ \( ( 1 - 2750900 T + 86837156595958 T^{2} - 2750900 p^{7} T^{3} + p^{14} T^{4} )^{2} \)
97$D_{4}$ \( ( 1 - 10775428 T + 106365922942022 T^{2} - 10775428 p^{7} T^{3} + p^{14} T^{4} )^{2} \)
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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

−7.86151162847529197433151104894, −7.50308585873780170413908115356, −7.46236174579479494275506119110, −6.87564748109282196154436680590, −6.81772042510681874500044321937, −6.24052699511575583314122410183, −6.23456275843139799314343726318, −6.03509611247963606863020043882, −5.70024300343179631559786528574, −5.12285849816972703870364537686, −5.06219223472678913098674798919, −4.92696570685865349414500650456, −4.57535092934554815865945964339, −4.32282292703971111593294853030, −3.84644080015949707032120265063, −3.66845952863941382051028974817, −3.40030123109960454889868790413, −3.14192765637467729629260568743, −2.72681849633106616245033763143, −2.59107403496330091305398323189, −2.52586582886278238204843425144, −1.69075014258348173834196100377, −1.41032234611579644247748232929, −1.08911330577338126458972500288, −0.943835904798295050894057429770, 0, 0, 0, 0, 0.943835904798295050894057429770, 1.08911330577338126458972500288, 1.41032234611579644247748232929, 1.69075014258348173834196100377, 2.52586582886278238204843425144, 2.59107403496330091305398323189, 2.72681849633106616245033763143, 3.14192765637467729629260568743, 3.40030123109960454889868790413, 3.66845952863941382051028974817, 3.84644080015949707032120265063, 4.32282292703971111593294853030, 4.57535092934554815865945964339, 4.92696570685865349414500650456, 5.06219223472678913098674798919, 5.12285849816972703870364537686, 5.70024300343179631559786528574, 6.03509611247963606863020043882, 6.23456275843139799314343726318, 6.24052699511575583314122410183, 6.81772042510681874500044321937, 6.87564748109282196154436680590, 7.46236174579479494275506119110, 7.50308585873780170413908115356, 7.86151162847529197433151104894

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