Properties

Label 8-1960e4-1.1-c1e4-0-1
Degree $8$
Conductor $1.476\times 10^{13}$
Sign $1$
Analytic cond. $59997.4$
Root an. cond. $3.95609$
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  − 3-s − 2·5-s − 2·9-s − 7·11-s − 6·13-s + 2·15-s + 5·17-s + 2·19-s − 2·23-s + 25-s + 13·27-s − 6·29-s − 16·31-s + 7·33-s + 4·37-s + 6·39-s − 4·41-s − 12·43-s + 4·45-s + 3·47-s − 5·51-s − 10·53-s + 14·55-s − 2·57-s + 16·59-s + 6·61-s + 12·65-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.894·5-s − 2/3·9-s − 2.11·11-s − 1.66·13-s + 0.516·15-s + 1.21·17-s + 0.458·19-s − 0.417·23-s + 1/5·25-s + 2.50·27-s − 1.11·29-s − 2.87·31-s + 1.21·33-s + 0.657·37-s + 0.960·39-s − 0.624·41-s − 1.82·43-s + 0.596·45-s + 0.437·47-s − 0.700·51-s − 1.37·53-s + 1.88·55-s − 0.264·57-s + 2.08·59-s + 0.768·61-s + 1.48·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{12} \cdot 5^{4} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(59997.4\)
Root analytic conductor: \(3.95609\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1960} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{12} \cdot 5^{4} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.08653824714\)
\(L(\frac12)\) \(\approx\) \(0.08653824714\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_2$ \( ( 1 + T + T^{2} )^{2} \)
7 \( 1 \)
good3$C_2$$\times$$C_2^2$ \( ( 1 + T + p T^{2} )^{2}( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} ) \)
11$D_4\times C_2$ \( 1 + 7 T + 23 T^{2} + 28 T^{3} + 16 T^{4} + 28 p T^{5} + 23 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \)
13$D_{4}$ \( ( 1 + 3 T + 20 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \)
17$D_4\times C_2$ \( 1 - 5 T - 7 T^{2} + 10 T^{3} + 310 T^{4} + 10 p T^{5} - 7 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \)
19$D_4\times C_2$ \( 1 - 2 T - 2 T^{2} + 64 T^{3} - 401 T^{4} + 64 p T^{5} - 2 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
23$D_4\times C_2$ \( 1 + 2 T - 10 T^{2} - 64 T^{3} - 425 T^{4} - 64 p T^{5} - 10 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
29$D_{4}$ \( ( 1 + 3 T + 52 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
41$D_{4}$ \( ( 1 + 2 T + 50 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
43$D_{4}$ \( ( 1 + 6 T + 62 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
47$D_4\times C_2$ \( 1 - 3 T - 13 T^{2} + 216 T^{3} - 2148 T^{4} + 216 p T^{5} - 13 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
53$D_4\times C_2$ \( 1 + 10 T + 2 T^{2} - 80 T^{3} + 1495 T^{4} - 80 p T^{5} + 2 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2^2$ \( ( 1 - 8 T + 5 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
61$D_4\times C_2$ \( 1 - 6 T - 62 T^{2} + 144 T^{3} + 3687 T^{4} + 144 p T^{5} - 62 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2^2$ \( ( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{4} \)
73$C_2^2$ \( ( 1 + 6 T - 37 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
79$D_4\times C_2$ \( 1 + 13 T + 43 T^{2} - 416 T^{3} - 3716 T^{4} - 416 p T^{5} + 43 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} \)
83$D_{4}$ \( ( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
89$D_4\times C_2$ \( 1 - 18 T + 98 T^{2} - 864 T^{3} + 14319 T^{4} - 864 p T^{5} + 98 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \)
97$D_{4}$ \( ( 1 + 9 T + 8 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \)
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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

−6.63838007274827768976215120909, −6.54420558530753452109757902959, −5.83896194245961385831454220607, −5.78277628407017144262564761831, −5.55982920964673505647032346025, −5.41772956969159163240487509344, −5.37743009397858160976185771918, −5.09811374328587672172980611004, −4.81955494592249660746089355664, −4.80794718594724459790544525855, −4.52986288982570213952608998396, −3.95768735116446882395191947979, −3.86405640080623690961331648619, −3.80217679927608474643589321611, −3.32030632819133422393339113903, −3.16440919245860644300137388453, −2.96981367599648077768162487046, −2.69146215536414717680184845843, −2.32378574808601135840992128001, −2.29075178515381734345710954658, −1.68676619215798766813347802765, −1.67734642223121590465169944242, −0.895390357547019583593448271302, −0.57363677831968160140908279823, −0.087368622420857707569264725285, 0.087368622420857707569264725285, 0.57363677831968160140908279823, 0.895390357547019583593448271302, 1.67734642223121590465169944242, 1.68676619215798766813347802765, 2.29075178515381734345710954658, 2.32378574808601135840992128001, 2.69146215536414717680184845843, 2.96981367599648077768162487046, 3.16440919245860644300137388453, 3.32030632819133422393339113903, 3.80217679927608474643589321611, 3.86405640080623690961331648619, 3.95768735116446882395191947979, 4.52986288982570213952608998396, 4.80794718594724459790544525855, 4.81955494592249660746089355664, 5.09811374328587672172980611004, 5.37743009397858160976185771918, 5.41772956969159163240487509344, 5.55982920964673505647032346025, 5.78277628407017144262564761831, 5.83896194245961385831454220607, 6.54420558530753452109757902959, 6.63838007274827768976215120909

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