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Properties

Label 8-48e4-1.1-c14e4-0-4
Degree $8$
Conductor $5308416$
Sign $1$
Analytic cond. $1.26839\times 10^{7}$
Root an. cond. $7.72514$
Motivic weight $14$
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  + 1.83e4·5-s − 3.18e6·9-s − 1.58e8·13-s + 2.62e8·17-s − 2.03e10·25-s − 5.99e10·29-s + 1.47e10·37-s + 4.46e11·41-s − 5.85e10·45-s + 1.72e12·49-s + 8.37e11·53-s − 3.82e12·61-s − 2.90e12·65-s − 8.96e12·73-s + 7.62e12·81-s + 4.81e12·85-s − 7.92e13·89-s − 8.16e13·97-s + 2.68e13·101-s − 1.42e14·109-s − 3.61e14·113-s + 5.05e14·117-s + 6.94e14·121-s − 4.53e14·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  + 0.235·5-s − 2/3·9-s − 2.52·13-s + 0.639·17-s − 3.34·25-s − 3.47·29-s + 0.155·37-s + 2.29·41-s − 0.156·45-s + 2.54·49-s + 0.712·53-s − 1.21·61-s − 0.593·65-s − 0.811·73-s + 1/3·81-s + 0.150·85-s − 1.79·89-s − 1.01·97-s + 0.250·101-s − 0.781·109-s − 1.53·113-s + 1.68·117-s + 1.82·121-s − 0.950·125-s − 0.816·145-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s+7)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(5308416\)    =    \(2^{16} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(1.26839\times 10^{7}\)
Root analytic conductor: \(7.72514\)
Motivic weight: \(14\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 5308416,\ (\ :7, 7, 7, 7),\ 1)\)

Particular Values

\(L(\frac{15}{2})\) \(\approx\) \(0.2855567862\)
\(L(\frac12)\) \(\approx\) \(0.2855567862\)
\(L(8)\) 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$C_2$ \( ( 1 + p^{13} T^{2} )^{2} \)
good5$D_{4}$ \( ( 1 - 1836 p T + 2064116062 p T^{2} - 1836 p^{15} T^{3} + p^{28} T^{4} )^{2} \)
7$D_4\times C_2$ \( 1 - 246225787516 p T^{2} + \)\(29\!\cdots\!02\)\( p^{2} T^{4} - 246225787516 p^{29} T^{6} + p^{56} T^{8} \)
11$D_4\times C_2$ \( 1 - 694873469355460 T^{2} + \)\(32\!\cdots\!82\)\( p^{2} T^{4} - 694873469355460 p^{28} T^{6} + p^{56} T^{8} \)
13$D_{4}$ \( ( 1 + 468604 p^{2} T + 14894253545478 p^{2} T^{2} + 468604 p^{16} T^{3} + p^{28} T^{4} )^{2} \)
17$D_{4}$ \( ( 1 - 131248404 T + 89556859180618022 T^{2} - 131248404 p^{14} T^{3} + p^{28} T^{4} )^{2} \)
19$D_4\times C_2$ \( 1 - 266206698601530628 T^{2} + \)\(91\!\cdots\!38\)\( T^{4} - 266206698601530628 p^{28} T^{6} + p^{56} T^{8} \)
23$D_4\times C_2$ \( 1 - 18669353752109478340 T^{2} + \)\(26\!\cdots\!02\)\( T^{4} - 18669353752109478340 p^{28} T^{6} + p^{56} T^{8} \)
29$D_{4}$ \( ( 1 + 29970024084 T + \)\(76\!\cdots\!86\)\( T^{2} + 29970024084 p^{14} T^{3} + p^{28} T^{4} )^{2} \)
31$D_4\times C_2$ \( 1 - \)\(28\!\cdots\!44\)\( T^{2} + \)\(32\!\cdots\!66\)\( T^{4} - \)\(28\!\cdots\!44\)\( p^{28} T^{6} + p^{56} T^{8} \)
37$D_{4}$ \( ( 1 - 7363936660 T + \)\(15\!\cdots\!18\)\( T^{2} - 7363936660 p^{14} T^{3} + p^{28} T^{4} )^{2} \)
41$D_{4}$ \( ( 1 - 223165175220 T + \)\(74\!\cdots\!22\)\( T^{2} - 223165175220 p^{14} T^{3} + p^{28} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - \)\(27\!\cdots\!00\)\( T^{2} + \)\(29\!\cdots\!42\)\( T^{4} - \)\(27\!\cdots\!00\)\( p^{28} T^{6} + p^{56} T^{8} \)
47$D_4\times C_2$ \( 1 - \)\(93\!\cdots\!52\)\( T^{2} + \)\(34\!\cdots\!38\)\( T^{4} - \)\(93\!\cdots\!52\)\( p^{28} T^{6} + p^{56} T^{8} \)
53$D_{4}$ \( ( 1 - 418505585868 T + \)\(27\!\cdots\!34\)\( T^{2} - 418505585868 p^{14} T^{3} + p^{28} T^{4} )^{2} \)
59$D_4\times C_2$ \( 1 - \)\(20\!\cdots\!00\)\( T^{2} + \)\(42\!\cdots\!02\)\( T^{4} - \)\(20\!\cdots\!00\)\( p^{28} T^{6} + p^{56} T^{8} \)
61$D_{4}$ \( ( 1 + 1914366463532 T + \)\(20\!\cdots\!98\)\( T^{2} + 1914366463532 p^{14} T^{3} + p^{28} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 - \)\(60\!\cdots\!32\)\( T^{2} + \)\(27\!\cdots\!98\)\( T^{4} - \)\(60\!\cdots\!32\)\( p^{28} T^{6} + p^{56} T^{8} \)
71$D_4\times C_2$ \( 1 - \)\(53\!\cdots\!80\)\( T^{2} + \)\(68\!\cdots\!62\)\( T^{4} - \)\(53\!\cdots\!80\)\( p^{28} T^{6} + p^{56} T^{8} \)
73$D_{4}$ \( ( 1 + 4480042803484 T + \)\(23\!\cdots\!22\)\( T^{2} + 4480042803484 p^{14} T^{3} + p^{28} T^{4} )^{2} \)
79$D_4\times C_2$ \( 1 + \)\(14\!\cdots\!92\)\( T^{2} + \)\(26\!\cdots\!98\)\( T^{4} + \)\(14\!\cdots\!92\)\( p^{28} T^{6} + p^{56} T^{8} \)
83$D_4\times C_2$ \( 1 - \)\(13\!\cdots\!40\)\( T^{2} - \)\(81\!\cdots\!78\)\( T^{4} - \)\(13\!\cdots\!40\)\( p^{28} T^{6} + p^{56} T^{8} \)
89$D_{4}$ \( ( 1 + 39648355746588 T + \)\(20\!\cdots\!58\)\( T^{2} + 39648355746588 p^{14} T^{3} + p^{28} T^{4} )^{2} \)
97$D_{4}$ \( ( 1 + 40846980584188 T + \)\(10\!\cdots\!74\)\( T^{2} + 40846980584188 p^{14} T^{3} + p^{28} T^{4} )^{2} \)
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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

−8.832303591196485442741353722822, −8.149420511343609181804906765456, −7.73930619614218649657045425620, −7.60576250257741363084021417498, −7.60095478199303356664827546332, −7.12435193946844931277864491633, −6.89283905572499657532508931028, −6.15126654393830711983331351351, −5.77975321697016729374914816904, −5.68435953337099099017734252201, −5.57976690180581823431831523900, −5.15237237488258813267848928621, −4.59185192939115062867672294386, −4.17364068296107408006648078477, −4.02232738632361431258895587701, −3.65253738303183294089785276585, −3.22645657012066066963293025958, −2.62526613317394808570143919227, −2.40089698574720847974529898266, −2.15923644257256158912401516200, −1.96565757500341278637422274864, −1.27097615346986426791915840014, −1.07443206013005774378417759891, −0.21129822697052535679676975595, −0.17225060090843901737809453024, 0.17225060090843901737809453024, 0.21129822697052535679676975595, 1.07443206013005774378417759891, 1.27097615346986426791915840014, 1.96565757500341278637422274864, 2.15923644257256158912401516200, 2.40089698574720847974529898266, 2.62526613317394808570143919227, 3.22645657012066066963293025958, 3.65253738303183294089785276585, 4.02232738632361431258895587701, 4.17364068296107408006648078477, 4.59185192939115062867672294386, 5.15237237488258813267848928621, 5.57976690180581823431831523900, 5.68435953337099099017734252201, 5.77975321697016729374914816904, 6.15126654393830711983331351351, 6.89283905572499657532508931028, 7.12435193946844931277864491633, 7.60095478199303356664827546332, 7.60576250257741363084021417498, 7.73930619614218649657045425620, 8.149420511343609181804906765456, 8.832303591196485442741353722822

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