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Properties

Label 8-93e4-1.1-c0e4-0-0
Degree $8$
Conductor $74805201$
Sign $1$
Analytic cond. $4.64045\times 10^{-6}$
Root an. cond. $0.215436$
Motivic weight $0$
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  − 3-s − 4-s − 2·7-s + 12-s − 2·13-s − 2·19-s + 2·21-s + 4·25-s + 2·28-s − 31-s − 2·37-s + 2·39-s + 3·43-s + 49-s + 2·52-s + 2·57-s − 2·61-s − 2·67-s − 2·73-s − 4·75-s + 2·76-s + 3·79-s − 2·84-s + 4·91-s + 93-s + 3·97-s − 4·100-s + ⋯
L(s)  = 1  − 3-s − 4-s − 2·7-s + 12-s − 2·13-s − 2·19-s + 2·21-s + 4·25-s + 2·28-s − 31-s − 2·37-s + 2·39-s + 3·43-s + 49-s + 2·52-s + 2·57-s − 2·61-s − 2·67-s − 2·73-s − 4·75-s + 2·76-s + 3·79-s − 2·84-s + 4·91-s + 93-s + 3·97-s − 4·100-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74805201 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74805201 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(74805201\)    =    \(3^{4} \cdot 31^{4}\)
Sign: $1$
Analytic conductor: \(4.64045\times 10^{-6}\)
Root analytic conductor: \(0.215436\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 74805201,\ (\ :0, 0, 0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.05420552636\)
\(L(\frac12)\) \(\approx\) \(0.05420552636\)
\(L(1)\) 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$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
31$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
good2$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
5$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
7$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
11$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
13$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
17$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
19$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
23$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
29$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
37$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
41$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
43$C_1$$\times$$C_4$ \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \)
47$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
53$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
59$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
61$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
67$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
71$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
73$C_4$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
79$C_1$$\times$$C_4$ \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \)
83$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
89$C_4$$\times$$C_4$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \)
97$C_1$$\times$$C_4$ \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \)
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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

−10.48743218973064051649932970661, −10.36797757186280028933745907370, −10.25229223046040944365644243583, −10.03532261497939277822132745853, −9.383994457173858562044692441485, −9.045382814740218732932819683367, −9.045062745149317170813775751523, −8.838152206804644912996589040524, −8.771530581454348144572623938258, −7.959656930620755612021368932808, −7.49029318966349959013723291821, −7.34947433847167642766250330443, −7.05804596571881808973016571167, −6.60718084941818390355171763636, −6.41943666264396079973304475783, −5.95144901391539568473880923115, −5.91979281389483345475406376034, −5.22099923391254890010555720139, −4.77436710369613119308281463036, −4.63388359943371166320155636122, −4.54016193188011225163629372491, −3.52379163668022342736277136955, −3.38399562972915395412111180289, −2.73902604091271210242942096147, −2.26669821585936102709468412872, 2.26669821585936102709468412872, 2.73902604091271210242942096147, 3.38399562972915395412111180289, 3.52379163668022342736277136955, 4.54016193188011225163629372491, 4.63388359943371166320155636122, 4.77436710369613119308281463036, 5.22099923391254890010555720139, 5.91979281389483345475406376034, 5.95144901391539568473880923115, 6.41943666264396079973304475783, 6.60718084941818390355171763636, 7.05804596571881808973016571167, 7.34947433847167642766250330443, 7.49029318966349959013723291821, 7.959656930620755612021368932808, 8.771530581454348144572623938258, 8.838152206804644912996589040524, 9.045062745149317170813775751523, 9.045382814740218732932819683367, 9.383994457173858562044692441485, 10.03532261497939277822132745853, 10.25229223046040944365644243583, 10.36797757186280028933745907370, 10.48743218973064051649932970661

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