Properties

Label 8-825e4-1.1-c1e4-0-5
Degree $8$
Conductor $463250390625$
Sign $1$
Analytic cond. $1883.32$
Root an. cond. $2.56664$
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  + 2·4-s − 2·9-s − 4·11-s − 5·16-s − 8·19-s − 16·31-s − 4·36-s − 8·44-s + 20·49-s + 8·61-s − 20·64-s − 16·76-s + 40·79-s + 3·81-s + 24·89-s + 8·99-s + 40·109-s + 10·121-s − 32·124-s + 127-s + 131-s + 137-s + 139-s + 10·144-s + 149-s + 151-s + 157-s + ⋯
L(s)  = 1  + 4-s − 2/3·9-s − 1.20·11-s − 5/4·16-s − 1.83·19-s − 2.87·31-s − 2/3·36-s − 1.20·44-s + 20/7·49-s + 1.02·61-s − 5/2·64-s − 1.83·76-s + 4.50·79-s + 1/3·81-s + 2.54·89-s + 0.804·99-s + 3.83·109-s + 0.909·121-s − 2.87·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 5/6·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{4} \cdot 5^{8} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(1883.32\)
Root analytic conductor: \(2.56664\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{825} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{4} \cdot 5^{8} \cdot 11^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.213608702\)
\(L(\frac12)\) \(\approx\) \(2.213608702\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 + T^{2} )^{2} \)
5 \( 1 \)
11$C_1$ \( ( 1 + T )^{4} \)
good2$C_2^2$ \( ( 1 - T^{2} + p^{2} T^{4} )^{2} \)
7$C_2^2$ \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \)
13$D_4\times C_2$ \( 1 - 20 T^{2} + 246 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \)
17$C_2$ \( ( 1 - p T^{2} )^{4} \)
19$D_{4}$ \( ( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 + 46 T^{2} + p^{2} T^{4} )^{2} \)
31$D_{4}$ \( ( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
37$D_4\times C_2$ \( 1 - 44 T^{2} + 2454 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \)
41$C_2^2$ \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - 68 T^{2} + 4086 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} \)
47$C_2^2$ \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \)
53$D_4\times C_2$ \( 1 - 44 T^{2} - 810 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \)
59$C_2^2$ \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{4} \)
67$C_2^2$ \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 - 68 T^{2} + 10086 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} \)
79$D_{4}$ \( ( 1 - 20 T + 246 T^{2} - 20 p T^{3} + p^{2} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - 20 T^{2} + 6966 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \)
89$D_{4}$ \( ( 1 - 12 T + 166 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 - 94 T^{2} + p^{2} T^{4} )^{2} \)
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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

−7.25393223053615710133423615004, −7.17526807701395907895401843232, −7.16872743778026337025899664539, −6.51541456726270564207280694401, −6.35331260975208968034684003138, −6.21277014476787892315552786071, −6.18818961983607569270190317593, −5.65801488458165075733864724970, −5.52905644023793588322019549537, −5.25391769033307639550965740407, −5.05252376397257103360975154502, −4.72282799358739013135706777104, −4.59422013344895188649664883340, −4.10690806310176834170250719206, −3.90649165950298740211252882117, −3.74981209447660614358341621758, −3.29695416041477541306898307707, −3.06403052853626780636676153200, −2.74653804274963272692888021960, −2.22015955346548991070108857088, −2.12230771397485103184706496613, −2.05173378084296385092226761997, −1.82204295442602736010174637171, −0.62720990351345168937278189652, −0.52660378947635185750152915647, 0.52660378947635185750152915647, 0.62720990351345168937278189652, 1.82204295442602736010174637171, 2.05173378084296385092226761997, 2.12230771397485103184706496613, 2.22015955346548991070108857088, 2.74653804274963272692888021960, 3.06403052853626780636676153200, 3.29695416041477541306898307707, 3.74981209447660614358341621758, 3.90649165950298740211252882117, 4.10690806310176834170250719206, 4.59422013344895188649664883340, 4.72282799358739013135706777104, 5.05252376397257103360975154502, 5.25391769033307639550965740407, 5.52905644023793588322019549537, 5.65801488458165075733864724970, 6.18818961983607569270190317593, 6.21277014476787892315552786071, 6.35331260975208968034684003138, 6.51541456726270564207280694401, 7.16872743778026337025899664539, 7.17526807701395907895401843232, 7.25393223053615710133423615004

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