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Properties

Label 8-539e4-1.1-c3e4-0-0
Degree $8$
Conductor $84402451441$
Sign $1$
Analytic cond. $1.02286\times 10^{6}$
Root an. cond. $5.63932$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 14·3-s − 4-s − 10·5-s + 28·6-s + 4·8-s + 82·9-s + 20·10-s − 44·11-s + 14·12-s − 58·13-s + 140·15-s − 51·16-s − 4·17-s − 164·18-s − 258·19-s + 10·20-s + 88·22-s + 8·23-s − 56·24-s − 160·25-s + 116·26-s − 250·27-s − 396·29-s − 280·30-s + 56·31-s + 154·32-s + ⋯
L(s)  = 1  − 0.707·2-s − 2.69·3-s − 1/8·4-s − 0.894·5-s + 1.90·6-s + 0.176·8-s + 3.03·9-s + 0.632·10-s − 1.20·11-s + 0.336·12-s − 1.23·13-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 + 0.852·22-s + 0.0725·23-s − 0.476·24-s − 1.27·25-s + 0.874·26-s − 1.78·27-s − 2.53·29-s − 1.70·30-s + 0.324·31-s + 0.850·32-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(7^{8} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(1.02286\times 10^{6}\)
Root analytic conductor: \(5.63932\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 7^{8} \cdot 11^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 \)
11$C_1$ \( ( 1 + p T )^{4} \)
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

−7.892434205765564065386499948064, −7.66141695559602901058534659429, −7.48077104042567469591784621353, −6.86650328361838932684795967218, −6.82397727298087141259821211633, −6.73100608269957684695051604806, −6.52050037129876300025702671199, −6.04102955223297609213079562793, −5.94518804516740200253043566473, −5.49773603614173117459254845016, −5.40437862728335527339831346095, −5.26151180245208531377533433192, −5.25545328422220610096555500904, −4.40889410361046725918048808818, −4.36430517548942854379778601609, −4.30551314122724776551806868186, −4.05487967977970461865797613183, −3.63152738026641187498420130147, −3.27751091892255259992883603749, −2.58049135459312609771529425922, −2.29851406344578814534564306992, −2.19068774123170297755583941703, −1.95910253702569987459652346742, −1.06911956426648270682099251308, −0.78620168823730262012030385833, 0, 0, 0, 0, 0.78620168823730262012030385833, 1.06911956426648270682099251308, 1.95910253702569987459652346742, 2.19068774123170297755583941703, 2.29851406344578814534564306992, 2.58049135459312609771529425922, 3.27751091892255259992883603749, 3.63152738026641187498420130147, 4.05487967977970461865797613183, 4.30551314122724776551806868186, 4.36430517548942854379778601609, 4.40889410361046725918048808818, 5.25545328422220610096555500904, 5.26151180245208531377533433192, 5.40437862728335527339831346095, 5.49773603614173117459254845016, 5.94518804516740200253043566473, 6.04102955223297609213079562793, 6.52050037129876300025702671199, 6.73100608269957684695051604806, 6.82397727298087141259821211633, 6.86650328361838932684795967218, 7.48077104042567469591784621353, 7.66141695559602901058534659429, 7.892434205765564065386499948064

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