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Properties

Label 8-242e4-1.1-c3e4-0-14
Degree $8$
Conductor $3429742096$
Sign $1$
Analytic cond. $41564.8$
Root an. cond. $3.77868$
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  + 8·2-s + 4·3-s + 40·4-s + 25·5-s + 32·6-s + 3·7-s + 160·8-s + 5·9-s + 200·10-s + 160·12-s − 41·13-s + 24·14-s + 100·15-s + 560·16-s + 52·17-s + 40·18-s + 16·19-s + 1.00e3·20-s + 12·21-s + 314·23-s + 640·24-s + 52·25-s − 328·26-s + 58·27-s + 120·28-s + 561·29-s + 800·30-s + ⋯
L(s)  = 1  + 2.82·2-s + 0.769·3-s + 5·4-s + 2.23·5-s + 2.17·6-s + 0.161·7-s + 7.07·8-s + 5/27·9-s + 6.32·10-s + 3.84·12-s − 0.874·13-s + 0.458·14-s + 1.72·15-s + 35/4·16-s + 0.741·17-s + 0.523·18-s + 0.193·19-s + 11.1·20-s + 0.124·21-s + 2.84·23-s + 5.44·24-s + 0.415·25-s − 2.47·26-s + 0.413·27-s + 0.809·28-s + 3.59·29-s + 4.86·30-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 11^{8}\)
Sign: $1$
Analytic conductor: \(41564.8\)
Root analytic conductor: \(3.77868\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 11^{8} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(101.7240166\)
\(L(\frac12)\) \(\approx\) \(101.7240166\)
\(L(\frac{5}{2})\) 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)$
bad2$C_1$ \( ( 1 - p T )^{4} \)
11 \( 1 \)
good3$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 11 T^{2} - 82 T^{3} + 1315 T^{4} - 82 p^{3} T^{5} + 11 p^{6} T^{6} - 4 p^{9} T^{7} + p^{12} T^{8} \)
5$C_2 \wr C_2\wr C_2$ \( 1 - p^{2} T + 573 T^{2} - 1533 p T^{3} + 103716 T^{4} - 1533 p^{4} T^{5} + 573 p^{6} T^{6} - p^{11} T^{7} + p^{12} T^{8} \)
7$C_2 \wr C_2\wr C_2$ \( 1 - 3 T + 279 T^{2} - 1591 T^{3} + 251100 T^{4} - 1591 p^{3} T^{5} + 279 p^{6} T^{6} - 3 p^{9} T^{7} + p^{12} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 + 41 T + 5637 T^{2} + 241261 T^{3} + 15241940 T^{4} + 241261 p^{3} T^{5} + 5637 p^{6} T^{6} + 41 p^{9} T^{7} + p^{12} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 - 52 T + 16573 T^{2} - 591848 T^{3} + 112551705 T^{4} - 591848 p^{3} T^{5} + 16573 p^{6} T^{6} - 52 p^{9} T^{7} + p^{12} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 - 16 T + 14825 T^{2} - 500832 T^{3} + 111187933 T^{4} - 500832 p^{3} T^{5} + 14825 p^{6} T^{6} - 16 p^{9} T^{7} + p^{12} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 - 314 T + 61816 T^{2} - 8042722 T^{3} + 950275950 T^{4} - 8042722 p^{3} T^{5} + 61816 p^{6} T^{6} - 314 p^{9} T^{7} + p^{12} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 - 561 T + 200635 T^{2} - 47945667 T^{3} + 8699420208 T^{4} - 47945667 p^{3} T^{5} + 200635 p^{6} T^{6} - 561 p^{9} T^{7} + p^{12} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 - 199 T + 51813 T^{2} + 256853 T^{3} + 389222024 T^{4} + 256853 p^{3} T^{5} + 51813 p^{6} T^{6} - 199 p^{9} T^{7} + p^{12} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 - 357 T + 160529 T^{2} - 42781809 T^{3} + 12141822140 T^{4} - 42781809 p^{3} T^{5} + 160529 p^{6} T^{6} - 357 p^{9} T^{7} + p^{12} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 - 32 T + 194093 T^{2} - 3202364 T^{3} + 18031305245 T^{4} - 3202364 p^{3} T^{5} + 194093 p^{6} T^{6} - 32 p^{9} T^{7} + p^{12} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 + 721 T + 420117 T^{2} + 154459221 T^{3} + 51447883420 T^{4} + 154459221 p^{3} T^{5} + 420117 p^{6} T^{6} + 721 p^{9} T^{7} + p^{12} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 - 403 T + 357463 T^{2} - 121226987 T^{3} + 52982120160 T^{4} - 121226987 p^{3} T^{5} + 357463 p^{6} T^{6} - 403 p^{9} T^{7} + p^{12} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 + 133 T + 365887 T^{2} + 13770175 T^{3} + 65244311076 T^{4} + 13770175 p^{3} T^{5} + 365887 p^{6} T^{6} + 133 p^{9} T^{7} + p^{12} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 - 1016 T + 1078585 T^{2} - 639078152 T^{3} + 358860597453 T^{4} - 639078152 p^{3} T^{5} + 1078585 p^{6} T^{6} - 1016 p^{9} T^{7} + p^{12} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 + 919 T + 1093783 T^{2} + 606559837 T^{3} + 388669851044 T^{4} + 606559837 p^{3} T^{5} + 1093783 p^{6} T^{6} + 919 p^{9} T^{7} + p^{12} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 - 289 T + 686735 T^{2} - 243308709 T^{3} + 267199450216 T^{4} - 243308709 p^{3} T^{5} + 686735 p^{6} T^{6} - 289 p^{9} T^{7} + p^{12} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 + 1205 T + 1608131 T^{2} + 1210556495 T^{3} + 899763338876 T^{4} + 1210556495 p^{3} T^{5} + 1608131 p^{6} T^{6} + 1205 p^{9} T^{7} + p^{12} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 + 1234 T + 1514091 T^{2} + 1290056552 T^{3} + 893250631145 T^{4} + 1290056552 p^{3} T^{5} + 1514091 p^{6} T^{6} + 1234 p^{9} T^{7} + p^{12} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 + 603 T + 1688657 T^{2} + 808010451 T^{3} + 1205184633704 T^{4} + 808010451 p^{3} T^{5} + 1688657 p^{6} T^{6} + 603 p^{9} T^{7} + p^{12} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 - 1514 T + 2070047 T^{2} - 2046213464 T^{3} + 1753896205685 T^{4} - 2046213464 p^{3} T^{5} + 2070047 p^{6} T^{6} - 1514 p^{9} T^{7} + p^{12} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 + 1101 T + 2406895 T^{2} + 1914547747 T^{3} + 2334666485008 T^{4} + 1914547747 p^{3} T^{5} + 2406895 p^{6} T^{6} + 1101 p^{9} T^{7} + p^{12} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 - 2116 T + 3632073 T^{2} - 4516910940 T^{3} + 5180506457941 T^{4} - 4516910940 p^{3} T^{5} + 3632073 p^{6} T^{6} - 2116 p^{9} T^{7} + p^{12} T^{8} \)
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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

−8.407415100618256507863181749024, −7.78775151342035122360411254103, −7.66126121953437845413214663385, −7.66118829125068378826964541529, −7.01837806674432088795166209380, −6.76923875522082618361551891195, −6.59519590444013883792772962866, −6.41392862355810161848537177862, −6.10189478832302152512382953750, −5.84457870857156428599971339694, −5.51508065993927315354377472924, −5.19976993165233748485825845904, −5.12266871297615454111413185467, −4.62222355263919397457984179371, −4.60172343632328888877082899121, −4.25765319206792989318675068124, −3.80799644999122393573564559534, −3.05760074250229547463509930258, −2.94317461071781194278453744965, −2.92740406880386426868475535022, −2.67578560900912292113851842675, −1.94828328982834482207014141597, −1.89962747601108052568772170071, −1.08676844579049933797407530991, −1.05999928890979395020567088457, 1.05999928890979395020567088457, 1.08676844579049933797407530991, 1.89962747601108052568772170071, 1.94828328982834482207014141597, 2.67578560900912292113851842675, 2.92740406880386426868475535022, 2.94317461071781194278453744965, 3.05760074250229547463509930258, 3.80799644999122393573564559534, 4.25765319206792989318675068124, 4.60172343632328888877082899121, 4.62222355263919397457984179371, 5.12266871297615454111413185467, 5.19976993165233748485825845904, 5.51508065993927315354377472924, 5.84457870857156428599971339694, 6.10189478832302152512382953750, 6.41392862355810161848537177862, 6.59519590444013883792772962866, 6.76923875522082618361551891195, 7.01837806674432088795166209380, 7.66118829125068378826964541529, 7.66126121953437845413214663385, 7.78775151342035122360411254103, 8.407415100618256507863181749024

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