Properties

Label 8-12e8-1.1-c12e4-0-1
Degree $8$
Conductor $429981696$
Sign $1$
Analytic cond. $3.00070\times 10^{8}$
Root an. cond. $11.4723$
Motivic weight $12$
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  − 2.23e4·5-s + 8.90e6·13-s − 7.25e7·17-s + 1.21e8·25-s + 7.28e8·29-s − 2.72e9·37-s + 7.16e9·41-s + 4.73e10·49-s − 3.26e10·53-s − 1.30e11·61-s − 1.99e11·65-s − 1.70e11·73-s + 1.62e12·85-s − 2.01e12·89-s + 1.49e12·97-s + 1.78e11·101-s − 3.70e12·109-s − 1.95e12·113-s + 2.24e11·121-s − 2.52e12·125-s + 127-s + 131-s + 137-s + 139-s − 1.63e13·145-s + 149-s + 151-s + ⋯
L(s)  = 1  − 1.43·5-s + 1.84·13-s − 3.00·17-s + 0.497·25-s + 1.22·29-s − 1.06·37-s + 1.50·41-s + 3.42·49-s − 1.47·53-s − 2.53·61-s − 2.64·65-s − 1.12·73-s + 4.30·85-s − 4.04·89-s + 1.79·97-s + 0.168·101-s − 2.20·109-s − 0.939·113-s + 0.0716·121-s − 0.663·125-s − 1.75·145-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(13-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+6)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(3.00070\times 10^{8}\)
Root analytic conductor: \(11.4723\)
Motivic weight: \(12\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 3^{8} ,\ ( \ : 6, 6, 6, 6 ),\ 1 )\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(0.01429145942\)
\(L(\frac12)\) \(\approx\) \(0.01429145942\)
\(L(7)\) 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 \( 1 \)
3 \( 1 \)
good5$D_{4}$ \( ( 1 + 11196 T + 5091542 p^{2} T^{2} + 11196 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
7$D_4\times C_2$ \( 1 - 47395207588 T^{2} + 19089410985529124262 p^{2} T^{4} - 47395207588 p^{24} T^{6} + p^{48} T^{8} \)
11$D_4\times C_2$ \( 1 - 1858523140 p^{2} T^{2} - \)\(10\!\cdots\!98\)\( p^{4} T^{4} - 1858523140 p^{26} T^{6} + p^{48} T^{8} \)
13$D_{4}$ \( ( 1 - 4452980 T + 51426312946566 T^{2} - 4452980 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
17$D_{4}$ \( ( 1 + 36273780 T + 1481760659979686 T^{2} + 36273780 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
19$D_4\times C_2$ \( 1 - 5850927212613988 T^{2} + \)\(16\!\cdots\!78\)\( T^{4} - 5850927212613988 p^{24} T^{6} + p^{48} T^{8} \)
23$D_4\times C_2$ \( 1 - 61390384196685700 T^{2} + \)\(18\!\cdots\!82\)\( T^{4} - 61390384196685700 p^{24} T^{6} + p^{48} T^{8} \)
29$D_{4}$ \( ( 1 - 364205556 T + 599383288421216966 T^{2} - 364205556 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
31$D_4\times C_2$ \( 1 - 404370170843358244 T^{2} - \)\(33\!\cdots\!74\)\( T^{4} - 404370170843358244 p^{24} T^{6} + p^{48} T^{8} \)
37$D_{4}$ \( ( 1 + 1362286940 T - 1500398694640026138 T^{2} + 1362286940 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
41$D_{4}$ \( ( 1 - 3582060300 T + 39700473140168754662 T^{2} - 3582060300 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 + 10651747351856438300 T^{2} - \)\(19\!\cdots\!98\)\( T^{4} + 10651747351856438300 p^{24} T^{6} + p^{48} T^{8} \)
47$D_4\times C_2$ \( 1 - \)\(18\!\cdots\!88\)\( T^{2} + \)\(21\!\cdots\!98\)\( T^{4} - \)\(18\!\cdots\!88\)\( p^{24} T^{6} + p^{48} T^{8} \)
53$D_{4}$ \( ( 1 + 16309635660 T + \)\(10\!\cdots\!18\)\( T^{2} + 16309635660 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
59$D_4\times C_2$ \( 1 - \)\(46\!\cdots\!60\)\( T^{2} + \)\(54\!\cdots\!22\)\( T^{4} - \)\(46\!\cdots\!60\)\( p^{24} T^{6} + p^{48} T^{8} \)
61$D_{4}$ \( ( 1 + 65368120892 T + \)\(39\!\cdots\!58\)\( T^{2} + 65368120892 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 - \)\(12\!\cdots\!48\)\( T^{2} + \)\(75\!\cdots\!18\)\( T^{4} - \)\(12\!\cdots\!48\)\( p^{24} T^{6} + p^{48} T^{8} \)
71$D_4\times C_2$ \( 1 - \)\(96\!\cdots\!40\)\( T^{2} + \)\(91\!\cdots\!62\)\( T^{4} - \)\(96\!\cdots\!40\)\( p^{24} T^{6} + p^{48} T^{8} \)
73$D_{4}$ \( ( 1 + 85248724540 T + \)\(33\!\cdots\!66\)\( T^{2} + 85248724540 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
79$D_4\times C_2$ \( 1 - \)\(21\!\cdots\!48\)\( T^{2} + \)\(18\!\cdots\!38\)\( T^{4} - \)\(21\!\cdots\!48\)\( p^{24} T^{6} + p^{48} T^{8} \)
83$D_4\times C_2$ \( 1 - \)\(21\!\cdots\!80\)\( T^{2} + \)\(33\!\cdots\!42\)\( T^{4} - \)\(21\!\cdots\!80\)\( p^{24} T^{6} + p^{48} T^{8} \)
89$D_{4}$ \( ( 1 + 1006111194948 T + \)\(72\!\cdots\!18\)\( T^{2} + 1006111194948 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
97$D_{4}$ \( ( 1 - 745709404100 T + \)\(13\!\cdots\!78\)\( T^{2} - 745709404100 p^{12} T^{3} + p^{24} T^{4} )^{2} \)
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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

−7.29633040232392443647898312721, −7.05271043542396182510326989955, −7.04718812981834925136541328875, −6.26135546824912030744458347174, −6.19522702714411441391222837820, −6.16772072010799184949257132151, −5.94065294069143562779693229466, −5.24611130516681982985649177949, −4.93388577174635670151004897815, −4.80656828556202259166869539988, −4.38323443342695217987462706248, −4.18716403479464095145586898881, −3.85760362906002473293337533071, −3.66703146793398472491953910968, −3.66519028689400111530591845244, −2.88772313817143026291890551822, −2.62350767515982957531964988927, −2.42316815838465671857241637497, −2.35915648404895311092608798776, −1.46867961831364473836248358598, −1.42444840865528461135039951281, −1.15268029725002265520136437166, −0.927762164319832476666237228757, −0.14075679068123061089199330297, −0.04252827997148282842156729805, 0.04252827997148282842156729805, 0.14075679068123061089199330297, 0.927762164319832476666237228757, 1.15268029725002265520136437166, 1.42444840865528461135039951281, 1.46867961831364473836248358598, 2.35915648404895311092608798776, 2.42316815838465671857241637497, 2.62350767515982957531964988927, 2.88772313817143026291890551822, 3.66519028689400111530591845244, 3.66703146793398472491953910968, 3.85760362906002473293337533071, 4.18716403479464095145586898881, 4.38323443342695217987462706248, 4.80656828556202259166869539988, 4.93388577174635670151004897815, 5.24611130516681982985649177949, 5.94065294069143562779693229466, 6.16772072010799184949257132151, 6.19522702714411441391222837820, 6.26135546824912030744458347174, 7.04718812981834925136541328875, 7.05271043542396182510326989955, 7.29633040232392443647898312721

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