Properties

Label 8-450e4-1.1-c1e4-0-12
Degree $8$
Conductor $41006250000$
Sign $1$
Analytic cond. $166.708$
Root an. cond. $1.89559$
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-s + 6·9-s + 28·19-s − 12·29-s + 20·31-s + 6·36-s − 18·41-s − 10·49-s − 18·59-s + 8·61-s − 64-s + 24·71-s + 28·76-s + 4·79-s + 27·81-s − 60·89-s − 24·101-s − 8·109-s − 12·116-s + 22·121-s + 20·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯
L(s)  = 1  + 1/2·4-s + 2·9-s + 6.42·19-s − 2.22·29-s + 3.59·31-s + 36-s − 2.81·41-s − 1.42·49-s − 2.34·59-s + 1.02·61-s − 1/8·64-s + 2.84·71-s + 3.21·76-s + 0.450·79-s + 3·81-s − 6.35·89-s − 2.38·101-s − 0.766·109-s − 1.11·116-s + 2·121-s + 1.79·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{8} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(166.708\)
Root analytic conductor: \(1.89559\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{8} \cdot 5^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(4.614875329\)
\(L(\frac12)\) \(\approx\) \(4.614875329\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - T^{2} + T^{4} \)
3$C_2$ \( ( 1 - p T^{2} )^{2} \)
5 \( 1 \)
good7$C_2^3$ \( 1 + 10 T^{2} + 51 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8} \)
11$C_2^2$ \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
13$C_2^2$$\times$$C_2^2$ \( ( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} )( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} ) \)
17$C_2^2$ \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \)
19$C_2$ \( ( 1 - 7 T + p T^{2} )^{4} \)
23$C_2^2$ \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 - 10 T + 69 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \)
41$C_2^2$ \( ( 1 + 9 T + 40 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \)
43$C_2^3$ \( 1 + 85 T^{2} + 5376 T^{4} + 85 p^{2} T^{6} + p^{4} T^{8} \)
47$C_2^3$ \( 1 + 58 T^{2} + 1155 T^{4} + 58 p^{2} T^{6} + p^{4} T^{8} \)
53$C_2^2$ \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + 9 T + 22 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 - 4 T - 45 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
67$C_2^3$ \( 1 - 35 T^{2} - 3264 T^{4} - 35 p^{2} T^{6} + p^{4} T^{8} \)
71$C_2$ \( ( 1 - 6 T + p T^{2} )^{4} \)
73$C_2^2$ \( ( 1 - 145 T^{2} + p^{2} T^{4} )^{2} \)
79$C_2^2$ \( ( 1 - 2 T - 75 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
83$C_2^3$ \( 1 + 85 T^{2} + 336 T^{4} + 85 p^{2} T^{6} + p^{4} T^{8} \)
89$C_2$ \( ( 1 + 15 T + p T^{2} )^{4} \)
97$C_2^3$ \( 1 - 95 T^{2} - 384 T^{4} - 95 p^{2} T^{6} + p^{4} T^{8} \)
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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

−7.84305900251725705959706272948, −7.73600425056645330039520316328, −7.42477684527285694488093438504, −7.33564427438749264516136564431, −7.17675994840916348818904059201, −6.76794444576115552892732635682, −6.64907816144280825636784433537, −6.56616372723269755128472040618, −5.85113760994653355153170394460, −5.84966417172568131262316635849, −5.37999641563612276641408699400, −5.11307092580265411474401431449, −5.06874552614248846357706697687, −4.93061296537513794508308742642, −4.29754635849590239302748258023, −4.17442594640257296276046123673, −3.62396070854871163622441942319, −3.51888415117457173888519504380, −3.05150949865388807070070605412, −2.96885164544055880103375250322, −2.70140185017344583631982108111, −1.77510522188239312836898970310, −1.59986130675274359312721141226, −1.20675279248670343376122428811, −0.923050122212285611337908036899, 0.923050122212285611337908036899, 1.20675279248670343376122428811, 1.59986130675274359312721141226, 1.77510522188239312836898970310, 2.70140185017344583631982108111, 2.96885164544055880103375250322, 3.05150949865388807070070605412, 3.51888415117457173888519504380, 3.62396070854871163622441942319, 4.17442594640257296276046123673, 4.29754635849590239302748258023, 4.93061296537513794508308742642, 5.06874552614248846357706697687, 5.11307092580265411474401431449, 5.37999641563612276641408699400, 5.84966417172568131262316635849, 5.85113760994653355153170394460, 6.56616372723269755128472040618, 6.64907816144280825636784433537, 6.76794444576115552892732635682, 7.17675994840916348818904059201, 7.33564427438749264516136564431, 7.42477684527285694488093438504, 7.73600425056645330039520316328, 7.84305900251725705959706272948

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