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Properties

Label 8-99e4-1.1-c2e4-0-1
Degree $8$
Conductor $96059601$
Sign $1$
Analytic cond. $52.9518$
Root an. cond. $1.64242$
Motivic weight $2$
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  + 5·3-s − 8·4-s − 5-s + 9·9-s + 22·11-s − 40·12-s − 5·15-s + 16·16-s + 8·20-s − 70·23-s − 24·25-s + 10·27-s − 37·31-s + 110·33-s − 72·36-s + 50·37-s − 176·44-s − 9·45-s + 50·47-s + 80·48-s − 98·49-s + 140·53-s − 22·55-s + 107·59-s + 40·60-s + 128·64-s + 35·67-s + ⋯
L(s)  = 1  + 5/3·3-s − 2·4-s − 1/5·5-s + 9-s + 2·11-s − 3.33·12-s − 1/3·15-s + 16-s + 2/5·20-s − 3.04·23-s − 0.959·25-s + 0.370·27-s − 1.19·31-s + 10/3·33-s − 2·36-s + 1.35·37-s − 4·44-s − 1/5·45-s + 1.06·47-s + 5/3·48-s − 2·49-s + 2.64·53-s − 2/5·55-s + 1.81·59-s + 2/3·60-s + 2·64-s + 0.522·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 96059601 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 96059601 ^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(96059601\)    =    \(3^{8} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(52.9518\)
Root analytic conductor: \(1.64242\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 96059601,\ (\ :1, 1, 1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.619892149\)
\(L(\frac12)\) \(\approx\) \(1.619892149\)
\(L(2)\) 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$C_2^2$ \( 1 - 5 T + 16 T^{2} - 5 p^{2} T^{3} + p^{4} T^{4} \)
11$C_2$ \( ( 1 - p T + p^{2} T^{2} )^{2} \)
good2$C_2$ \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \)
5$C_2$$\times$$C_2^2$ \( ( 1 + T + p^{2} T^{2} )^{2}( 1 - T - 24 T^{2} - p^{2} T^{3} + p^{4} T^{4} ) \)
7$C_2$ \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \)
13$C_2$ \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \)
17$C_1$$\times$$C_1$ \( ( 1 - p T )^{4}( 1 + p T )^{4} \)
19$C_1$$\times$$C_1$ \( ( 1 - p T )^{4}( 1 + p T )^{4} \)
23$C_2^2$ \( ( 1 + 35 T + 696 T^{2} + 35 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
29$C_2$ \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \)
31$C_2$$\times$$C_2^2$ \( ( 1 + 37 T + p^{2} T^{2} )^{2}( 1 - 37 T + 408 T^{2} - 37 p^{2} T^{3} + p^{4} T^{4} ) \)
37$C_2^2$ \( ( 1 - 25 T - 744 T^{2} - 25 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
41$C_2$ \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \)
43$C_2$ \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \)
47$C_2$$\times$$C_2^2$ \( ( 1 - 50 T + p^{2} T^{2} )^{2}( 1 + 50 T + 291 T^{2} + 50 p^{2} T^{3} + p^{4} T^{4} ) \)
53$C_2^2$ \( ( 1 - 70 T + 2091 T^{2} - 70 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
59$C_2$$\times$$C_2^2$ \( ( 1 - 107 T + p^{2} T^{2} )^{2}( 1 + 107 T + 7968 T^{2} + 107 p^{2} T^{3} + p^{4} T^{4} ) \)
61$C_2$ \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \)
67$C_2$$\times$$C_2^2$ \( ( 1 - 35 T + p^{2} T^{2} )^{2}( 1 + 35 T - 3264 T^{2} + 35 p^{2} T^{3} + p^{4} T^{4} ) \)
71$C_2^2$ \( ( 1 - 133 T + 12648 T^{2} - 133 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
73$C_1$$\times$$C_1$ \( ( 1 - p T )^{4}( 1 + p T )^{4} \)
79$C_2$ \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \)
83$C_2$ \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \)
89$C_2$ \( ( 1 + 97 T + p^{2} T^{2} )^{4} \)
97$C_2$$\times$$C_2^2$ \( ( 1 - 95 T + p^{2} T^{2} )^{2}( 1 + 95 T - 384 T^{2} + 95 p^{2} T^{3} + p^{4} T^{4} ) \)
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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

−9.763107219660218626259808733902, −9.591489685871797049968206061889, −9.573524189804433755054133352698, −9.084262389611961774997975136614, −8.884496503802651411223212483739, −8.474547119626839421733139629869, −8.372555217585446257836424463522, −8.271666887359261913528393109967, −7.892550243933393159494460740447, −7.47290802530179845710408692352, −7.05031358750028301674340784193, −6.79354281243527887399447871529, −6.35199257482414031820063792884, −5.94987081560060856901505759975, −5.66461758017330791770345816113, −5.13284181754084849905187329076, −4.82093555565082250289760390825, −4.04631014888807816696410136593, −3.99749114779319185806101575996, −3.85846167388524849987433256439, −3.76399349076974344936036590563, −2.72542384397262436552264681440, −2.27129939785814048184041785537, −1.74037403272695330788016102573, −0.57763374904627582870481295596, 0.57763374904627582870481295596, 1.74037403272695330788016102573, 2.27129939785814048184041785537, 2.72542384397262436552264681440, 3.76399349076974344936036590563, 3.85846167388524849987433256439, 3.99749114779319185806101575996, 4.04631014888807816696410136593, 4.82093555565082250289760390825, 5.13284181754084849905187329076, 5.66461758017330791770345816113, 5.94987081560060856901505759975, 6.35199257482414031820063792884, 6.79354281243527887399447871529, 7.05031358750028301674340784193, 7.47290802530179845710408692352, 7.892550243933393159494460740447, 8.271666887359261913528393109967, 8.372555217585446257836424463522, 8.474547119626839421733139629869, 8.884496503802651411223212483739, 9.084262389611961774997975136614, 9.573524189804433755054133352698, 9.591489685871797049968206061889, 9.763107219660218626259808733902

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