Properties

Label 8-150e4-1.1-c1e4-0-2
Degree $8$
Conductor $506250000$
Sign $1$
Analytic cond. $2.05813$
Root an. cond. $1.09442$
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-s + 3-s + 5·5-s + 6-s + 8·7-s + 5·10-s − 2·11-s − 6·13-s + 8·14-s + 5·15-s − 12·17-s + 8·21-s − 2·22-s − 6·23-s + 10·25-s − 6·26-s + 5·30-s − 12·31-s − 32-s − 2·33-s − 12·34-s + 40·35-s + 13·37-s − 6·39-s − 12·41-s + 8·42-s + 4·43-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 2.23·5-s + 0.408·6-s + 3.02·7-s + 1.58·10-s − 0.603·11-s − 1.66·13-s + 2.13·14-s + 1.29·15-s − 2.91·17-s + 1.74·21-s − 0.426·22-s − 1.25·23-s + 2·25-s − 1.17·26-s + 0.912·30-s − 2.15·31-s − 0.176·32-s − 0.348·33-s − 2.05·34-s + 6.76·35-s + 2.13·37-s − 0.960·39-s − 1.87·41-s + 1.23·42-s + 0.609·43-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(2.05813\)
Root analytic conductor: \(1.09442\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{4} \cdot 5^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(3.507704728\)
\(L(\frac12)\) \(\approx\) \(3.507704728\)
\(L(\frac{3}{2})\) 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$C_4$ \( 1 - T + T^{2} - T^{3} + T^{4} \)
3$C_4$ \( 1 - T + T^{2} - T^{3} + T^{4} \)
5$C_4$ \( 1 - p T + 3 p T^{2} - p^{2} T^{3} + p^{2} T^{4} \)
good7$C_2$ \( ( 1 - 2 T + p T^{2} )^{4} \)
11$C_4\times C_2$ \( 1 + 2 T + 13 T^{2} + 34 T^{3} + 225 T^{4} + 34 p T^{5} + 13 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2^2:C_4$ \( 1 + 6 T + 23 T^{2} + 120 T^{3} + 601 T^{4} + 120 p T^{5} + 23 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2^2:C_4$ \( 1 + 12 T + 37 T^{2} - 180 T^{3} - 1619 T^{4} - 180 p T^{5} + 37 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2^2:C_4$ \( 1 + 21 T^{2} - 10 T^{3} + 381 T^{4} - 10 p T^{5} + 21 p^{2} T^{6} + p^{4} T^{8} \)
23$C_4\times C_2$ \( 1 + 6 T + 13 T^{2} - 60 T^{3} - 659 T^{4} - 60 p T^{5} + 13 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2^2:C_4$ \( 1 - 19 T^{2} + 120 T^{3} + 721 T^{4} + 120 p T^{5} - 19 p^{2} T^{6} + p^{4} T^{8} \)
31$C_2^2:C_4$ \( 1 + 12 T + 113 T^{2} + 834 T^{3} + 5605 T^{4} + 834 p T^{5} + 113 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2^2:C_4$ \( 1 - 13 T + 27 T^{2} + 445 T^{3} - 4264 T^{4} + 445 p T^{5} + 27 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2^2:C_4$ \( 1 + 12 T + 53 T^{2} + 444 T^{3} + 4405 T^{4} + 444 p T^{5} + 53 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
43$D_{4}$ \( ( 1 - 2 T + 82 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
47$C_2^2:C_4$ \( 1 + 2 T + 17 T^{2} - 130 T^{3} + 761 T^{4} - 130 p T^{5} + 17 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2^2:C_4$ \( 1 + 21 T + 253 T^{2} + 2535 T^{3} + 21196 T^{4} + 2535 p T^{5} + 253 p^{2} T^{6} + 21 p^{3} T^{7} + p^{4} T^{8} \)
59$C_4\times C_2$ \( 1 - 20 T + 181 T^{2} - 1600 T^{3} + 14601 T^{4} - 1600 p T^{5} + 181 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2^2:C_4$ \( 1 + 2 T + 3 T^{2} + 424 T^{3} + 4265 T^{4} + 424 p T^{5} + 3 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2^2:C_4$ \( 1 - 18 T + 77 T^{2} - 30 T^{3} + 961 T^{4} - 30 p T^{5} + 77 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \)
71$C_4\times C_2$ \( 1 - 28 T + 313 T^{2} - 2126 T^{3} + 14805 T^{4} - 2126 p T^{5} + 313 p^{2} T^{6} - 28 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2^2:C_4$ \( 1 - 14 T + 3 T^{2} + 980 T^{3} - 9259 T^{4} + 980 p T^{5} + 3 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \)
79$C_4\times C_2$ \( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} \)
83$C_4\times C_2$ \( 1 + 6 T - 47 T^{2} - 780 T^{3} - 779 T^{4} - 780 p T^{5} - 47 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2^2:C_4$ \( 1 + 5 T - 79 T^{2} - 5 p T^{3} + 5276 T^{4} - 5 p^{2} T^{5} - 79 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2^2:C_4$ \( 1 - 18 T + 47 T^{2} + 1740 T^{3} - 27059 T^{4} + 1740 p T^{5} + 47 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \)
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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

−9.687818995382257380037828020586, −9.243439402267130846148126510136, −8.967736304019538588441462549927, −8.881303827775368396817517740481, −8.499241886291921564136946711727, −8.095332281353903721876412967852, −7.76726723566014285915545147718, −7.76483694790135152754973126685, −7.64982664406823497799484814159, −6.86851847503758559561602606013, −6.52182546105273119245285082556, −6.36060986634045218351650888427, −6.31836009160644899111225545213, −5.31749180948017034830304535510, −5.28497309654397536401566236720, −5.18463910985769388165492696502, −4.98102964864389107454151966734, −4.56622908050813456726919366037, −4.21501948897060548742146284126, −3.88651645387973616330895132183, −3.16269405794208447701174193661, −2.39424812705518774258246690486, −2.13936383242117023427396859604, −1.89545622112054634888781633824, −1.89414489831281707489994724004, 1.89414489831281707489994724004, 1.89545622112054634888781633824, 2.13936383242117023427396859604, 2.39424812705518774258246690486, 3.16269405794208447701174193661, 3.88651645387973616330895132183, 4.21501948897060548742146284126, 4.56622908050813456726919366037, 4.98102964864389107454151966734, 5.18463910985769388165492696502, 5.28497309654397536401566236720, 5.31749180948017034830304535510, 6.31836009160644899111225545213, 6.36060986634045218351650888427, 6.52182546105273119245285082556, 6.86851847503758559561602606013, 7.64982664406823497799484814159, 7.76483694790135152754973126685, 7.76726723566014285915545147718, 8.095332281353903721876412967852, 8.499241886291921564136946711727, 8.881303827775368396817517740481, 8.967736304019538588441462549927, 9.243439402267130846148126510136, 9.687818995382257380037828020586

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