Properties

Label 8-825e4-1.1-c1e4-0-21
Degree $8$
Conductor $463250390625$
Sign $1$
Analytic cond. $1883.32$
Root an. cond. $2.56664$
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  + 3·2-s + 3-s + 7·4-s + 3·6-s + 3·7-s + 15·8-s + 9·11-s + 7·12-s + 9·13-s + 9·14-s + 30·16-s − 2·17-s − 10·19-s + 3·21-s + 27·22-s + 4·23-s + 15·24-s + 27·26-s + 21·28-s − 10·29-s + 8·31-s + 57·32-s + 9·33-s − 6·34-s + 3·37-s − 30·38-s + 9·39-s + ⋯
L(s)  = 1  + 2.12·2-s + 0.577·3-s + 7/2·4-s + 1.22·6-s + 1.13·7-s + 5.30·8-s + 2.71·11-s + 2.02·12-s + 2.49·13-s + 2.40·14-s + 15/2·16-s − 0.485·17-s − 2.29·19-s + 0.654·21-s + 5.75·22-s + 0.834·23-s + 3.06·24-s + 5.29·26-s + 3.96·28-s − 1.85·29-s + 1.43·31-s + 10.0·32-s + 1.56·33-s − 1.02·34-s + 0.493·37-s − 4.86·38-s + 1.44·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{4} \cdot 5^{8} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(1883.32\)
Root analytic conductor: \(2.56664\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{825} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{4} \cdot 5^{8} \cdot 11^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(40.75236032\)
\(L(\frac12)\) \(\approx\) \(40.75236032\)
\(L(\frac{3}{2})\) 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$C_4$ \( 1 - T + T^{2} - T^{3} + T^{4} \)
5 \( 1 \)
11$C_4$ \( 1 - 9 T + 41 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
good2$C_2^2:C_4$ \( 1 - 3 T + p T^{2} + T^{4} + p^{3} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
7$C_4\times C_2$ \( 1 - 3 T + 2 T^{2} + 15 T^{3} - 59 T^{4} + 15 p T^{5} + 2 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2^2:C_4$ \( 1 - 9 T + 18 T^{2} + 115 T^{3} - 789 T^{4} + 115 p T^{5} + 18 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2^2:C_4$ \( 1 + 2 T - 13 T^{2} + 20 T^{3} + 341 T^{4} + 20 p T^{5} - 13 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2^2:C_4$ \( 1 + 10 T + 21 T^{2} - 70 T^{3} - 469 T^{4} - 70 p T^{5} + 21 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
23$D_{4}$ \( ( 1 - 2 T + 27 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
29$C_4\times C_2$ \( 1 + 10 T + 31 T^{2} + 200 T^{3} + 1821 T^{4} + 200 p T^{5} + 31 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2^2:C_4$ \( 1 - 8 T + 3 T^{2} - 46 T^{3} + 1175 T^{4} - 46 p T^{5} + 3 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2^2:C_4$ \( 1 - 3 T - 18 T^{2} - 155 T^{3} + 1851 T^{4} - 155 p T^{5} - 18 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2^2:C_4$ \( 1 - 23 T + 208 T^{2} - 961 T^{3} + 3975 T^{4} - 961 p T^{5} + 208 p^{2} T^{6} - 23 p^{3} T^{7} + p^{4} T^{8} \)
43$D_{4}$ \( ( 1 + 8 T + 97 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
47$C_2^2:C_4$ \( 1 - 3 T - 43 T^{2} + 45 T^{3} + 2116 T^{4} + 45 p T^{5} - 43 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2^2:C_4$ \( 1 + 6 T + 23 T^{2} - 120 T^{3} - 1319 T^{4} - 120 p T^{5} + 23 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2^2:C_4$ \( 1 + 20 T + 131 T^{2} + 530 T^{3} + 3851 T^{4} + 530 p T^{5} + 131 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2^2:C_4$ \( 1 - 3 T - 7 T^{2} - 441 T^{3} + 4900 T^{4} - 441 p T^{5} - 7 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
67$D_{4}$ \( ( 1 + T + 33 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \)
71$C_2^2:C_4$ \( 1 + 27 T + 253 T^{2} + 819 T^{3} + 100 T^{4} + 819 p T^{5} + 253 p^{2} T^{6} + 27 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2^2:C_4$ \( 1 + 6 T - 57 T^{2} - 130 T^{3} + 4761 T^{4} - 130 p T^{5} - 57 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2^2:C_4$ \( 1 - 5 T + 6 T^{2} - 715 T^{3} + 9821 T^{4} - 715 p T^{5} + 6 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2^2:C_4$ \( 1 + 21 T + 88 T^{2} - 915 T^{3} - 13199 T^{4} - 915 p T^{5} + 88 p^{2} T^{6} + 21 p^{3} T^{7} + p^{4} T^{8} \)
89$D_{4}$ \( ( 1 - 10 T + 183 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \)
97$C_2^2:C_4$ \( 1 - 33 T + 537 T^{2} - 6655 T^{3} + 71196 T^{4} - 6655 p T^{5} + 537 p^{2} T^{6} - 33 p^{3} T^{7} + p^{4} T^{8} \)
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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

−7.33283314748248424864949418338, −6.90630198202972515988559466165, −6.76376119656564707927676876062, −6.61693588325579851170971818714, −6.17364899300002192912197226579, −6.06700034358211929452775101708, −5.96210959527830253077821517010, −5.89861139907497348068045792573, −5.81575795181926125409648590146, −4.86663565183004833878027929973, −4.78697101678138887233239429765, −4.69045219401515244383716389448, −4.54831882445941260423682106502, −4.12260353507927922030783595375, −3.98288775432142664922663179286, −3.74445650967726680849714243566, −3.49920101415938192584510478330, −3.40461081004274588252449003868, −2.80195510001472176980855940453, −2.60393170154822045121802688264, −2.22610273173337920982609058495, −1.87472168090155952906866400247, −1.35410848970383396818218321580, −1.32549324197738212600302607405, −1.23861158627981513011651957616, 1.23861158627981513011651957616, 1.32549324197738212600302607405, 1.35410848970383396818218321580, 1.87472168090155952906866400247, 2.22610273173337920982609058495, 2.60393170154822045121802688264, 2.80195510001472176980855940453, 3.40461081004274588252449003868, 3.49920101415938192584510478330, 3.74445650967726680849714243566, 3.98288775432142664922663179286, 4.12260353507927922030783595375, 4.54831882445941260423682106502, 4.69045219401515244383716389448, 4.78697101678138887233239429765, 4.86663565183004833878027929973, 5.81575795181926125409648590146, 5.89861139907497348068045792573, 5.96210959527830253077821517010, 6.06700034358211929452775101708, 6.17364899300002192912197226579, 6.61693588325579851170971818714, 6.76376119656564707927676876062, 6.90630198202972515988559466165, 7.33283314748248424864949418338

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