Properties

Label 8-21e8-1.1-c1e4-0-3
Degree $8$
Conductor $37822859361$
Sign $1$
Analytic cond. $153.766$
Root an. cond. $1.87654$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 4·4-s + 4·16-s − 8·25-s − 4·37-s − 4·43-s − 16·64-s + 44·67-s + 20·79-s − 32·100-s − 4·109-s + 40·121-s + 127-s + 131-s + 137-s + 139-s − 16·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s − 16·172-s + 173-s + 179-s + 181-s + 191-s + ⋯
L(s)  = 1  + 2·4-s + 16-s − 8/5·25-s − 0.657·37-s − 0.609·43-s − 2·64-s + 5.37·67-s + 2.25·79-s − 3.19·100-s − 0.383·109-s + 3.63·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1.31·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.153·169-s − 1.21·172-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{8} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(153.766\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{441} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{8} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(3.259717519\)
\(L(\frac12)\) \(\approx\) \(3.259717519\)
\(L(\frac{3}{2})\) 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)$
bad3 \( 1 \)
7 \( 1 \)
good2$C_2^2$ \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
5$C_2^2$ \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )^{2}( 1 + 5 T + p T^{2} )^{2} \)
17$C_2^2$ \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 - 35 T^{2} + p^{2} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2$ \( ( 1 - 11 T + p T^{2} )^{2}( 1 + 11 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + T + p T^{2} )^{4} \)
41$C_2^2$ \( ( 1 + 28 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2$ \( ( 1 + T + p T^{2} )^{4} \)
47$C_2^2$ \( ( 1 - 56 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - 98 T^{2} + p^{2} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + 94 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{2} \)
67$C_2$ \( ( 1 - 11 T + p T^{2} )^{4} \)
71$C_2^2$ \( ( 1 - 92 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2$ \( ( 1 - 17 T + p T^{2} )^{2}( 1 + 17 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 5 T + p T^{2} )^{4} \)
83$C_2^2$ \( ( 1 + 112 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 + 154 T^{2} + p^{2} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 - 86 T^{2} + p^{2} T^{4} )^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.032555063951711344137063250386, −7.67560926915879912079886185546, −7.58365130693717008466942491144, −7.12166478652107226530339109384, −7.06990121167278531388781701873, −6.95162811633682740881105368789, −6.58075505144410173824752364124, −6.29147651339654169986309928524, −6.15771216219854371217483104082, −5.96535343522107660029560353359, −5.65366490005888899785754801000, −5.20584833639502952382641569862, −5.02908713655860041858033502440, −4.82912548424657138651501699310, −4.44389664720983271010962041782, −3.95544712116567946792405562081, −3.66194779668353224752148952161, −3.51952462200294649916587556444, −3.25260139742650708884155216828, −2.48901394909076953594804391716, −2.46678554053813862554345726233, −2.20586491300823955334880921721, −1.82784642471843825147680567386, −1.42032966318660902566473438361, −0.59536133950747632232977552167, 0.59536133950747632232977552167, 1.42032966318660902566473438361, 1.82784642471843825147680567386, 2.20586491300823955334880921721, 2.46678554053813862554345726233, 2.48901394909076953594804391716, 3.25260139742650708884155216828, 3.51952462200294649916587556444, 3.66194779668353224752148952161, 3.95544712116567946792405562081, 4.44389664720983271010962041782, 4.82912548424657138651501699310, 5.02908713655860041858033502440, 5.20584833639502952382641569862, 5.65366490005888899785754801000, 5.96535343522107660029560353359, 6.15771216219854371217483104082, 6.29147651339654169986309928524, 6.58075505144410173824752364124, 6.95162811633682740881105368789, 7.06990121167278531388781701873, 7.12166478652107226530339109384, 7.58365130693717008466942491144, 7.67560926915879912079886185546, 8.032555063951711344137063250386

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