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Properties

Label 8-847e4-1.1-c3e4-0-0
Degree $8$
Conductor $514675673281$
Sign $1$
Analytic cond. $6.23731\times 10^{6}$
Root an. cond. $7.06927$
Motivic weight $3$
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  + 2·2-s + 14·3-s − 4-s + 10·5-s + 28·6-s + 28·7-s − 4·8-s + 82·9-s + 20·10-s − 14·12-s − 58·13-s + 56·14-s + 140·15-s − 51·16-s − 4·17-s + 164·18-s − 258·19-s − 10·20-s + 392·21-s + 8·23-s − 56·24-s − 160·25-s − 116·26-s + 250·27-s − 28·28-s + 396·29-s + 280·30-s + ⋯
L(s)  = 1  + 0.707·2-s + 2.69·3-s − 1/8·4-s + 0.894·5-s + 1.90·6-s + 1.51·7-s − 0.176·8-s + 3.03·9-s + 0.632·10-s − 0.336·12-s − 1.23·13-s + 1.06·14-s + 2.40·15-s − 0.796·16-s − 0.0570·17-s + 2.14·18-s − 3.11·19-s − 0.111·20-s + 4.07·21-s + 0.0725·23-s − 0.476·24-s − 1.27·25-s − 0.874·26-s + 1.78·27-s − 0.188·28-s + 2.53·29-s + 1.70·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(7^{4} \cdot 11^{8}\)
Sign: $1$
Analytic conductor: \(6.23731\times 10^{6}\)
Root analytic conductor: \(7.06927\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 7^{4} \cdot 11^{8} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(19.74089037\)
\(L(\frac12)\) \(\approx\) \(19.74089037\)
\(L(\frac{5}{2})\) 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)$
bad7$C_1$ \( ( 1 - p T )^{4} \)
11 \( 1 \)
good2$C_2 \wr S_4$ \( 1 - p T + 5 T^{2} - p^{3} T^{3} + p^{6} T^{4} - p^{6} T^{5} + 5 p^{6} T^{6} - p^{10} T^{7} + p^{12} T^{8} \)
3$C_2 \wr S_4$ \( 1 - 14 T + 38 p T^{2} - 698 T^{3} + 3682 T^{4} - 698 p^{3} T^{5} + 38 p^{7} T^{6} - 14 p^{9} T^{7} + p^{12} T^{8} \)
5$C_2 \wr S_4$ \( 1 - 2 p T + 52 p T^{2} - 1102 T^{3} + 29734 T^{4} - 1102 p^{3} T^{5} + 52 p^{7} T^{6} - 2 p^{10} T^{7} + p^{12} T^{8} \)
13$C_2 \wr S_4$ \( 1 + 58 T + 3862 T^{2} + 39850 T^{3} + 2368354 T^{4} + 39850 p^{3} T^{5} + 3862 p^{6} T^{6} + 58 p^{9} T^{7} + p^{12} T^{8} \)
17$C_2 \wr S_4$ \( 1 + 4 T + 13466 T^{2} - 198500 T^{3} + 81336754 T^{4} - 198500 p^{3} T^{5} + 13466 p^{6} T^{6} + 4 p^{9} T^{7} + p^{12} T^{8} \)
19$C_2 \wr S_4$ \( 1 + 258 T + 2296 p T^{2} + 5147154 T^{3} + 489918366 T^{4} + 5147154 p^{3} T^{5} + 2296 p^{7} T^{6} + 258 p^{9} T^{7} + p^{12} T^{8} \)
23$C_2 \wr S_4$ \( 1 - 8 T + 26276 T^{2} + 1258456 T^{3} + 325878550 T^{4} + 1258456 p^{3} T^{5} + 26276 p^{6} T^{6} - 8 p^{9} T^{7} + p^{12} T^{8} \)
29$C_2 \wr S_4$ \( 1 - 396 T + 138428 T^{2} - 30598020 T^{3} + 5584930806 T^{4} - 30598020 p^{3} T^{5} + 138428 p^{6} T^{6} - 396 p^{9} T^{7} + p^{12} T^{8} \)
31$C_2 \wr S_4$ \( 1 + 56 T + 87274 T^{2} + 3513992 T^{3} + 3413701858 T^{4} + 3513992 p^{3} T^{5} + 87274 p^{6} T^{6} + 56 p^{9} T^{7} + p^{12} T^{8} \)
37$C_2 \wr S_4$ \( 1 - 84 T + 139096 T^{2} - 12117036 T^{3} + 8970963870 T^{4} - 12117036 p^{3} T^{5} + 139096 p^{6} T^{6} - 84 p^{9} T^{7} + p^{12} T^{8} \)
41$C_2 \wr S_4$ \( 1 + 52 T + 191978 T^{2} - 4000964 T^{3} + 16303189330 T^{4} - 4000964 p^{3} T^{5} + 191978 p^{6} T^{6} + 52 p^{9} T^{7} + p^{12} T^{8} \)
43$C_2 \wr S_4$ \( 1 + 408 T + 227224 T^{2} + 65489736 T^{3} + 24699468414 T^{4} + 65489736 p^{3} T^{5} + 227224 p^{6} T^{6} + 408 p^{9} T^{7} + p^{12} T^{8} \)
47$C_2 \wr S_4$ \( 1 - 8 T + 293642 T^{2} - 14787896 T^{3} + 39096224482 T^{4} - 14787896 p^{3} T^{5} + 293642 p^{6} T^{6} - 8 p^{9} T^{7} + p^{12} T^{8} \)
53$C_2 \wr S_4$ \( 1 - 624 T + 731456 T^{2} - 291124560 T^{3} + 173869150638 T^{4} - 291124560 p^{3} T^{5} + 731456 p^{6} T^{6} - 624 p^{9} T^{7} + p^{12} T^{8} \)
59$C_2 \wr S_4$ \( 1 + 238 T + 320090 T^{2} - 3740918 T^{3} + 64718281666 T^{4} - 3740918 p^{3} T^{5} + 320090 p^{6} T^{6} + 238 p^{9} T^{7} + p^{12} T^{8} \)
61$C_2 \wr S_4$ \( 1 - 162 T + 320206 T^{2} + 56573262 T^{3} + 48989538114 T^{4} + 56573262 p^{3} T^{5} + 320206 p^{6} T^{6} - 162 p^{9} T^{7} + p^{12} T^{8} \)
67$C_2 \wr S_4$ \( 1 - 20 p T + 1030948 T^{2} - 550745180 T^{3} + 298359795814 T^{4} - 550745180 p^{3} T^{5} + 1030948 p^{6} T^{6} - 20 p^{10} T^{7} + p^{12} T^{8} \)
71$C_2 \wr S_4$ \( 1 - 1788 T + 2193428 T^{2} - 1809946572 T^{3} + 1240921367286 T^{4} - 1809946572 p^{3} T^{5} + 2193428 p^{6} T^{6} - 1788 p^{9} T^{7} + p^{12} T^{8} \)
73$C_2 \wr S_4$ \( 1 + 1456 T + 1167226 T^{2} + 565357096 T^{3} + 283420832722 T^{4} + 565357096 p^{3} T^{5} + 1167226 p^{6} T^{6} + 1456 p^{9} T^{7} + p^{12} T^{8} \)
79$C_2 \wr S_4$ \( 1 - 1324 T + 2043976 T^{2} - 1611726940 T^{3} + 1469807561518 T^{4} - 1611726940 p^{3} T^{5} + 2043976 p^{6} T^{6} - 1324 p^{9} T^{7} + p^{12} T^{8} \)
83$C_2 \wr S_4$ \( 1 + 450 T + 1903832 T^{2} + 631295250 T^{3} + 1545243120894 T^{4} + 631295250 p^{3} T^{5} + 1903832 p^{6} T^{6} + 450 p^{9} T^{7} + p^{12} T^{8} \)
89$C_2 \wr S_4$ \( 1 + 3072 T + 5688284 T^{2} + 7201250304 T^{3} + 6916861622502 T^{4} + 7201250304 p^{3} T^{5} + 5688284 p^{6} T^{6} + 3072 p^{9} T^{7} + p^{12} T^{8} \)
97$C_2 \wr S_4$ \( 1 + 652 T + 1754332 T^{2} + 1157057716 T^{3} + 2404953539782 T^{4} + 1157057716 p^{3} T^{5} + 1754332 p^{6} T^{6} + 652 p^{9} T^{7} + p^{12} T^{8} \)
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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

−6.85499734329122203101798381548, −6.70667597428426725583791059024, −6.65834063268351398515170450865, −6.19087026861756053298857249678, −6.05355310086277650346268154438, −5.66151317764648870832740442579, −5.23748487651365023579615087764, −5.20006468332940115647370405781, −5.11961573762048942432391398474, −4.47912252751872200064768012615, −4.45398473129286304803071245715, −4.29988367753596253747982693244, −4.08032869886432579776275979658, −3.83635299478462737059996578924, −3.32745693309031466662825034179, −3.22276895315659848655754333007, −2.75232050615603012180054495817, −2.48715025033123340933215350948, −2.47104483377202402105291552707, −2.00032933383020723923421643628, −1.96975157668850305351477473474, −1.82996201855095756863721720762, −1.24961415042202564201619500976, −0.63312203665333486677318917132, −0.33210805545570359202129519817, 0.33210805545570359202129519817, 0.63312203665333486677318917132, 1.24961415042202564201619500976, 1.82996201855095756863721720762, 1.96975157668850305351477473474, 2.00032933383020723923421643628, 2.47104483377202402105291552707, 2.48715025033123340933215350948, 2.75232050615603012180054495817, 3.22276895315659848655754333007, 3.32745693309031466662825034179, 3.83635299478462737059996578924, 4.08032869886432579776275979658, 4.29988367753596253747982693244, 4.45398473129286304803071245715, 4.47912252751872200064768012615, 5.11961573762048942432391398474, 5.20006468332940115647370405781, 5.23748487651365023579615087764, 5.66151317764648870832740442579, 6.05355310086277650346268154438, 6.19087026861756053298857249678, 6.65834063268351398515170450865, 6.70667597428426725583791059024, 6.85499734329122203101798381548

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