Properties

Label 8-273e4-1.1-c1e4-0-8
Degree $8$
Conductor $5554571841$
Sign $1$
Analytic cond. $22.5818$
Root an. cond. $1.47645$
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  + 6·9-s + 4·16-s + 2·19-s − 8·31-s − 20·37-s + 36·43-s − 11·49-s − 22·67-s + 14·73-s + 27·81-s + 10·97-s − 12·103-s + 38·109-s + 127-s + 131-s + 137-s + 139-s + 24·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + 12·171-s + 173-s + 179-s + ⋯
L(s)  = 1  + 2·9-s + 16-s + 0.458·19-s − 1.43·31-s − 3.28·37-s + 5.48·43-s − 1.57·49-s − 2.68·67-s + 1.63·73-s + 3·81-s + 1.01·97-s − 1.18·103-s + 3.63·109-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 2·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-s + 0.917·171-s + 0.0760·173-s + 0.0747·179-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{4} \cdot 7^{4} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(22.5818\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{273} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{4} \cdot 7^{4} \cdot 13^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.351247120\)
\(L(\frac12)\) \(\approx\) \(2.351247120\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - p T^{2} )^{2} \)
7$C_2^2$ \( 1 + 11 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
good2$C_2^2$$\times$$C_2^2$ \( ( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} )( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} ) \)
5$C_2^3$ \( 1 - p^{2} T^{4} + p^{4} T^{8} \)
11$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
19$C_2$$\times$$C_2^2$ \( ( 1 - T + p T^{2} )^{2}( 1 - 37 T^{2} + p^{2} T^{4} ) \)
23$C_2^2$ \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \)
31$C_2$$\times$$C_2^2$ \( ( 1 + 4 T + p T^{2} )^{2}( 1 + 59 T^{2} + p^{2} T^{4} ) \)
37$C_2$$\times$$C_2^2$ \( ( 1 + 10 T + p T^{2} )^{2}( 1 - 73 T^{2} + p^{2} T^{4} ) \)
41$C_2^3$ \( 1 - p^{2} T^{4} + p^{4} T^{8} \)
43$C_2$ \( ( 1 - 13 T + p T^{2} )^{2}( 1 - 5 T + p T^{2} )^{2} \)
47$C_2^3$ \( 1 - p^{2} T^{4} + p^{4} T^{8} \)
53$C_2^2$ \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \)
59$C_2^3$ \( 1 - p^{2} T^{4} + p^{4} T^{8} \)
61$C_2^2$ \( ( 1 - 121 T^{2} + p^{2} T^{4} )^{2} \)
67$C_2$$\times$$C_2^2$ \( ( 1 + 11 T + p T^{2} )^{2}( 1 - 13 T^{2} + p^{2} T^{4} ) \)
71$C_2^3$ \( 1 - p^{2} T^{4} + p^{4} T^{8} \)
73$C_2$$\times$$C_2^2$ \( ( 1 - 7 T + p T^{2} )^{2}( 1 + 143 T^{2} + p^{2} T^{4} ) \)
79$C_2^2$$\times$$C_2^2$ \( ( 1 - 142 T^{2} + p^{2} T^{4} )( 1 + 131 T^{2} + p^{2} T^{4} ) \)
83$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
89$C_2^3$ \( 1 - p^{2} T^{4} + p^{4} T^{8} \)
97$C_2$$\times$$C_2^2$ \( ( 1 - 5 T + p T^{2} )^{2}( 1 + 2 T^{2} + p^{2} T^{4} ) \)
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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

−9.000823340604805562211047877385, −8.365634259790819036733879139932, −8.009143662998636406389413906742, −7.77312291924538248562853475793, −7.67429731731366436108446736691, −7.43749487051906280962738954880, −6.99716998511362283849988254925, −6.96147991899138037417452249385, −6.83762142493620906227815171043, −6.17161410218688959431182686228, −6.03815345921504085971535402317, −5.64738131824238056118633075569, −5.54543726325998252814720280880, −5.18031596331125900166035076671, −4.62029792875819667985293043945, −4.60554796710661485156932875889, −4.26228655679691115691827820720, −3.80423190773104554637021019618, −3.57770513903349666813018849959, −3.29239212165727272561433624926, −2.89508173716522223620062954783, −2.07529505677522911963470350920, −2.01087686788100944455718525059, −1.39961560219702737429592142104, −0.851910396260609784028907903638, 0.851910396260609784028907903638, 1.39961560219702737429592142104, 2.01087686788100944455718525059, 2.07529505677522911963470350920, 2.89508173716522223620062954783, 3.29239212165727272561433624926, 3.57770513903349666813018849959, 3.80423190773104554637021019618, 4.26228655679691115691827820720, 4.60554796710661485156932875889, 4.62029792875819667985293043945, 5.18031596331125900166035076671, 5.54543726325998252814720280880, 5.64738131824238056118633075569, 6.03815345921504085971535402317, 6.17161410218688959431182686228, 6.83762142493620906227815171043, 6.96147991899138037417452249385, 6.99716998511362283849988254925, 7.43749487051906280962738954880, 7.67429731731366436108446736691, 7.77312291924538248562853475793, 8.009143662998636406389413906742, 8.365634259790819036733879139932, 9.000823340604805562211047877385

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