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Properties

Label 8-2178e4-1.1-c3e4-0-4
Degree $8$
Conductor $2.250\times 10^{13}$
Sign $1$
Analytic cond. $2.72706\times 10^{8}$
Root an. cond. $11.3360$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 8·2-s + 40·4-s + 6·5-s − 12·7-s + 160·8-s + 48·10-s − 12·13-s − 96·14-s + 560·16-s + 108·17-s − 192·19-s + 240·20-s − 156·23-s − 89·25-s − 96·26-s − 480·28-s + 408·29-s − 26·31-s + 1.79e3·32-s + 864·34-s − 72·35-s − 224·37-s − 1.53e3·38-s + 960·40-s + 348·41-s − 540·43-s − 1.24e3·46-s + ⋯
L(s)  = 1  + 2.82·2-s + 5·4-s + 0.536·5-s − 0.647·7-s + 7.07·8-s + 1.51·10-s − 0.256·13-s − 1.83·14-s + 35/4·16-s + 1.54·17-s − 2.31·19-s + 2.68·20-s − 1.41·23-s − 0.711·25-s − 0.724·26-s − 3.23·28-s + 2.61·29-s − 0.150·31-s + 9.89·32-s + 4.35·34-s − 0.347·35-s − 0.995·37-s − 6.55·38-s + 3.79·40-s + 1.32·41-s − 1.91·43-s − 4.00·46-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{8} \cdot 11^{8}\)
Sign: $1$
Analytic conductor: \(2.72706\times 10^{8}\)
Root analytic conductor: \(11.3360\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{8} \cdot 11^{8} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(58.70850470\)
\(L(\frac12)\) \(\approx\) \(58.70850470\)
\(L(\frac{5}{2})\) 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$C_1$ \( ( 1 - p T )^{4} \)
3 \( 1 \)
11 \( 1 \)
good5$C_2 \wr C_2\wr C_2$ \( 1 - 6 T + p^{3} T^{2} - 2682 T^{3} + 17424 T^{4} - 2682 p^{3} T^{5} + p^{9} T^{6} - 6 p^{9} T^{7} + p^{12} T^{8} \)
7$C_2 \wr C_2\wr C_2$ \( 1 + 12 T + 421 T^{2} - 1584 T^{3} + 22872 T^{4} - 1584 p^{3} T^{5} + 421 p^{6} T^{6} + 12 p^{9} T^{7} + p^{12} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 + 12 T + 3613 T^{2} - 145116 T^{3} + 3654096 T^{4} - 145116 p^{3} T^{5} + 3613 p^{6} T^{6} + 12 p^{9} T^{7} + p^{12} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 - 108 T + 14981 T^{2} - 965844 T^{3} + 87217512 T^{4} - 965844 p^{3} T^{5} + 14981 p^{6} T^{6} - 108 p^{9} T^{7} + p^{12} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 + 192 T + 36241 T^{2} + 3946824 T^{3} + 399943380 T^{4} + 3946824 p^{3} T^{5} + 36241 p^{6} T^{6} + 192 p^{9} T^{7} + p^{12} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 + 156 T + 26324 T^{2} + 3657852 T^{3} + 521573382 T^{4} + 3657852 p^{3} T^{5} + 26324 p^{6} T^{6} + 156 p^{9} T^{7} + p^{12} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 - 408 T + 154013 T^{2} - 32947416 T^{3} + 6379323540 T^{4} - 32947416 p^{3} T^{5} + 154013 p^{6} T^{6} - 408 p^{9} T^{7} + p^{12} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 + 26 T + 63973 T^{2} + 2713394 T^{3} + 2585839300 T^{4} + 2713394 p^{3} T^{5} + 63973 p^{6} T^{6} + 26 p^{9} T^{7} + p^{12} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 + 224 T + 120934 T^{2} + 14498840 T^{3} + 6427931839 T^{4} + 14498840 p^{3} T^{5} + 120934 p^{6} T^{6} + 224 p^{9} T^{7} + p^{12} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 - 348 T + 188825 T^{2} - 28234476 T^{3} + 12556837572 T^{4} - 28234476 p^{3} T^{5} + 188825 p^{6} T^{6} - 348 p^{9} T^{7} + p^{12} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 + 540 T + 154084 T^{2} + 6722028 T^{3} - 3630542250 T^{4} + 6722028 p^{3} T^{5} + 154084 p^{6} T^{6} + 540 p^{9} T^{7} + p^{12} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 + 132 T - 5584 T^{2} - 2814444 T^{3} + 11742434142 T^{4} - 2814444 p^{3} T^{5} - 5584 p^{6} T^{6} + 132 p^{9} T^{7} + p^{12} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 - 1470 T + 1218857 T^{2} - 707251086 T^{3} + 311764150428 T^{4} - 707251086 p^{3} T^{5} + 1218857 p^{6} T^{6} - 1470 p^{9} T^{7} + p^{12} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 - 684 T + 587348 T^{2} - 361830204 T^{3} + 170284744566 T^{4} - 361830204 p^{3} T^{5} + 587348 p^{6} T^{6} - 684 p^{9} T^{7} + p^{12} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 + 1320 T + 1352113 T^{2} + 940531968 T^{3} + 511934208600 T^{4} + 940531968 p^{3} T^{5} + 1352113 p^{6} T^{6} + 1320 p^{9} T^{7} + p^{12} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 + 530 T + 370585 T^{2} - 24331534 T^{3} + 2099978008 T^{4} - 24331534 p^{3} T^{5} + 370585 p^{6} T^{6} + 530 p^{9} T^{7} + p^{12} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 + 936 T + 1123856 T^{2} + 887790312 T^{3} + 552383921214 T^{4} + 887790312 p^{3} T^{5} + 1123856 p^{6} T^{6} + 936 p^{9} T^{7} + p^{12} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 - 2352 T + 3201001 T^{2} - 3027668256 T^{3} + 2176722989040 T^{4} - 3027668256 p^{3} T^{5} + 3201001 p^{6} T^{6} - 2352 p^{9} T^{7} + p^{12} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 - 192 T + 1027897 T^{2} + 268153704 T^{3} + 447334282380 T^{4} + 268153704 p^{3} T^{5} + 1027897 p^{6} T^{6} - 192 p^{9} T^{7} + p^{12} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 - 3192 T + 5737424 T^{2} - 6882238584 T^{3} + 6059716449774 T^{4} - 6882238584 p^{3} T^{5} + 5737424 p^{6} T^{6} - 3192 p^{9} T^{7} + p^{12} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 + 726 T + 2159513 T^{2} + 1355072922 T^{3} + 2088276182784 T^{4} + 1355072922 p^{3} T^{5} + 2159513 p^{6} T^{6} + 726 p^{9} T^{7} + p^{12} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 + 100 T + 1742626 T^{2} + 220181056 T^{3} + 1926353653675 T^{4} + 220181056 p^{3} T^{5} + 1742626 p^{6} T^{6} + 100 p^{9} T^{7} + p^{12} T^{8} \)
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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

−6.14274306307168487067485469627, −5.79141745684306557585954807227, −5.71697656192231967498004134136, −5.42772979895962140387613619302, −5.18836247918977376291102930867, −4.86132909359743100676963816102, −4.83011958007529503878640861571, −4.75003058563330138121114127614, −4.33186960371914191786257075650, −4.05339956831246500180957576684, −3.98029536539203699748126678461, −3.79513073864759310168071128597, −3.52420247135962004407948485114, −3.32768433181205484027406455871, −3.00703679308425228806087536874, −2.80524112393093618961241078448, −2.73834350744249528029734102700, −2.16627021772031361543235629345, −2.01417557974931697776465168295, −1.93549462037410864529380543538, −1.89076304602179756610704907313, −1.12758976590568674289198657108, −1.02764830296674240301676179834, −0.52672594948006315758672326571, −0.35874924889588405707698964391, 0.35874924889588405707698964391, 0.52672594948006315758672326571, 1.02764830296674240301676179834, 1.12758976590568674289198657108, 1.89076304602179756610704907313, 1.93549462037410864529380543538, 2.01417557974931697776465168295, 2.16627021772031361543235629345, 2.73834350744249528029734102700, 2.80524112393093618961241078448, 3.00703679308425228806087536874, 3.32768433181205484027406455871, 3.52420247135962004407948485114, 3.79513073864759310168071128597, 3.98029536539203699748126678461, 4.05339956831246500180957576684, 4.33186960371914191786257075650, 4.75003058563330138121114127614, 4.83011958007529503878640861571, 4.86132909359743100676963816102, 5.18836247918977376291102930867, 5.42772979895962140387613619302, 5.71697656192231967498004134136, 5.79141745684306557585954807227, 6.14274306307168487067485469627

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