Properties

Label 8-4608e4-1.1-c1e4-0-33
Degree $8$
Conductor $4.509\times 10^{14}$
Sign $1$
Analytic cond. $1.83298\times 10^{6}$
Root an. cond. $6.06589$
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  + 4·5-s + 12·13-s + 16·17-s + 8·25-s + 4·29-s + 12·37-s + 12·49-s − 20·53-s − 4·61-s + 48·65-s + 64·85-s + 64·97-s + 44·101-s − 12·109-s − 72·113-s + 28·125-s + 127-s + 131-s + 137-s + 139-s + 16·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 72·169-s + ⋯
L(s)  = 1  + 1.78·5-s + 3.32·13-s + 3.88·17-s + 8/5·25-s + 0.742·29-s + 1.97·37-s + 12/7·49-s − 2.74·53-s − 0.512·61-s + 5.95·65-s + 6.94·85-s + 6.49·97-s + 4.37·101-s − 1.14·109-s − 6.77·113-s + 2.50·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.32·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 5.53·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{36} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(1.83298\times 10^{6}\)
Root analytic conductor: \(6.06589\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{4608} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{36} \cdot 3^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(26.02245174\)
\(L(\frac12)\) \(\approx\) \(26.02245174\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 2 T + p T^{2} )^{2} \)
7$C_2^2$ \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
17$C_2$ \( ( 1 - 4 T + p T^{2} )^{4} \)
19$C_2^3$ \( 1 - 46 T^{4} + p^{4} T^{8} \)
23$C_2^2$ \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 + 54 T^{2} + p^{2} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 - 66 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2$ \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2} \)
59$C_2^3$ \( 1 - 6286 T^{4} + p^{4} T^{8} \)
61$C_2$ \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \)
67$C_2^3$ \( 1 + 4946 T^{4} + p^{4} T^{8} \)
71$C_2^2$ \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \)
79$C_2^2$ \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{2} \)
83$C_2^3$ \( 1 - 5678 T^{4} + p^{4} T^{8} \)
89$C_2^2$ \( ( 1 - 142 T^{2} + p^{2} T^{4} )^{2} \)
97$C_2$ \( ( 1 - 16 T + p T^{2} )^{4} \)
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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

−5.98744867663181494478613199548, −5.61195525274651024987528710319, −5.57840344345731255775798798352, −5.43738958023535851727039908601, −5.32909346551212528587484424485, −4.80579747490104092016538662075, −4.73157634080827542054283438549, −4.61495633903138081475506032159, −4.33802654249764818531529143622, −3.91785674939952230494127664973, −3.74776781978794530087447607966, −3.73759686176021479365313475180, −3.38479871937764520245286781257, −3.22324899175836172551946218979, −2.94567712307032330549647425733, −2.92203226714161181329521378041, −2.73836065319098562026578838415, −2.08776848751191137247460606318, −1.99649641577090893055883672527, −1.76981248079616675041469287305, −1.53045489176891164453057196729, −1.23150278463987248618962966705, −0.937060742615307666943291868182, −0.78869548348634475447640652247, −0.69952977551712132652291930951, 0.69952977551712132652291930951, 0.78869548348634475447640652247, 0.937060742615307666943291868182, 1.23150278463987248618962966705, 1.53045489176891164453057196729, 1.76981248079616675041469287305, 1.99649641577090893055883672527, 2.08776848751191137247460606318, 2.73836065319098562026578838415, 2.92203226714161181329521378041, 2.94567712307032330549647425733, 3.22324899175836172551946218979, 3.38479871937764520245286781257, 3.73759686176021479365313475180, 3.74776781978794530087447607966, 3.91785674939952230494127664973, 4.33802654249764818531529143622, 4.61495633903138081475506032159, 4.73157634080827542054283438549, 4.80579747490104092016538662075, 5.32909346551212528587484424485, 5.43738958023535851727039908601, 5.57840344345731255775798798352, 5.61195525274651024987528710319, 5.98744867663181494478613199548

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