Properties

Label 8-1232e4-1.1-c3e4-0-4
Degree $8$
Conductor $2.304\times 10^{12}$
Sign $1$
Analytic cond. $2.79194\times 10^{7}$
Root an. cond. $8.52586$
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  − 14·3-s + 10·5-s + 28·7-s + 82·9-s + 44·11-s + 58·13-s − 140·15-s + 4·17-s − 258·19-s − 392·21-s − 8·23-s − 160·25-s − 250·27-s − 396·29-s + 56·31-s − 616·33-s + 280·35-s + 84·37-s − 812·39-s + 52·41-s − 408·43-s + 820·45-s − 8·47-s + 490·49-s − 56·51-s + 624·53-s + 440·55-s + ⋯
L(s)  = 1  − 2.69·3-s + 0.894·5-s + 1.51·7-s + 3.03·9-s + 1.20·11-s + 1.23·13-s − 2.40·15-s + 0.0570·17-s − 3.11·19-s − 4.07·21-s − 0.0725·23-s − 1.27·25-s − 1.78·27-s − 2.53·29-s + 0.324·31-s − 3.24·33-s + 1.35·35-s + 0.373·37-s − 3.33·39-s + 0.198·41-s − 1.44·43-s + 2.71·45-s − 0.0248·47-s + 10/7·49-s − 0.153·51-s + 1.61·53-s + 1.07·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 7^{4} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(2.79194\times 10^{7}\)
Root analytic conductor: \(8.52586\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1232} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{16} \cdot 7^{4} \cdot 11^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

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)$
bad2 \( 1 \)
7$C_1$ \( ( 1 - p T )^{4} \)
11$C_1$ \( ( 1 - p T )^{4} \)
good3$C_2 \wr S_4$ \( 1 + 14 T + 38 p T^{2} + 698 T^{3} + 3682 T^{4} + 698 p^{3} T^{5} + 38 p^{7} T^{6} + 14 p^{9} T^{7} + p^{12} T^{8} \)
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

−6.71143609886022583311250348305, −6.69933273507036299691978238529, −6.37590973688619449848919834087, −6.24664216259771538233751562328, −6.08191703891784493496339746517, −5.82315993035821251812876127539, −5.72583120814007604219502350474, −5.51448642916635854182502317376, −5.44761148624790900441710460375, −4.97986559769831821620795053815, −4.87025595922762375451523719895, −4.68583120707649640975149808911, −4.23548561772427339989478905749, −4.06157890309676685964490331456, −3.89004983930361509780981077114, −3.73679660251847105435503951111, −3.67309679932694440419065489878, −2.80655585984836223741936477997, −2.51842369260886060443881656025, −2.22255664627633354396678006536, −2.20028472804444197017009096140, −1.46557721086066685720925114035, −1.41131365284591403608802996092, −1.30457310507710787551504086594, −1.16552832259182026736239862710, 0, 0, 0, 0, 1.16552832259182026736239862710, 1.30457310507710787551504086594, 1.41131365284591403608802996092, 1.46557721086066685720925114035, 2.20028472804444197017009096140, 2.22255664627633354396678006536, 2.51842369260886060443881656025, 2.80655585984836223741936477997, 3.67309679932694440419065489878, 3.73679660251847105435503951111, 3.89004983930361509780981077114, 4.06157890309676685964490331456, 4.23548561772427339989478905749, 4.68583120707649640975149808911, 4.87025595922762375451523719895, 4.97986559769831821620795053815, 5.44761148624790900441710460375, 5.51448642916635854182502317376, 5.72583120814007604219502350474, 5.82315993035821251812876127539, 6.08191703891784493496339746517, 6.24664216259771538233751562328, 6.37590973688619449848919834087, 6.69933273507036299691978238529, 6.71143609886022583311250348305

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