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Properties

Label 8-72e4-1.1-c6e4-0-0
Degree $8$
Conductor $26873856$
Sign $1$
Analytic cond. $75274.8$
Root an. cond. $4.06987$
Motivic weight $6$
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  − 432·7-s − 4.35e3·13-s − 2.69e4·19-s − 2.35e4·25-s − 5.61e4·31-s − 2.37e5·37-s + 2.84e3·43-s − 1.86e5·49-s + 1.46e5·61-s + 1.54e6·67-s + 2.81e5·73-s + 1.25e6·79-s + 1.88e6·91-s − 1.32e6·97-s − 3.92e6·103-s − 3.04e6·109-s + 2.98e6·121-s + 127-s + 131-s + 1.16e7·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  − 1.25·7-s − 1.98·13-s − 3.92·19-s − 1.50·25-s − 1.88·31-s − 4.67·37-s + 0.0358·43-s − 1.58·49-s + 0.646·61-s + 5.14·67-s + 0.724·73-s + 2.55·79-s + 2.49·91-s − 1.44·97-s − 3.59·103-s − 2.35·109-s + 1.68·121-s + 4.94·133-s + 0.148·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 26873856 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26873856 ^{s/2} \, \Gamma_{\C}(s+3)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(26873856\)    =    \(2^{12} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(75274.8\)
Root analytic conductor: \(4.06987\)
Motivic weight: \(6\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 26873856,\ (\ :3, 3, 3, 3),\ 1)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.004520314638\)
\(L(\frac12)\) \(\approx\) \(0.004520314638\)
\(L(4)\) 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$D_4\times C_2$ \( 1 + 4704 p T^{2} + 16711154 p^{2} T^{4} + 4704 p^{13} T^{6} + p^{24} T^{8} \)
7$D_{4}$ \( ( 1 + 216 T + 163442 T^{2} + 216 p^{6} T^{3} + p^{12} T^{4} )^{2} \)
11$D_4\times C_2$ \( 1 - 271020 p T^{2} + 5196482322662 T^{4} - 271020 p^{13} T^{6} + p^{24} T^{8} \)
13$D_{4}$ \( ( 1 + 2176 T + 6744882 T^{2} + 2176 p^{6} T^{3} + p^{12} T^{4} )^{2} \)
17$D_4\times C_2$ \( 1 - 5032320 p T^{2} + 2991357595693442 T^{4} - 5032320 p^{13} T^{6} + p^{24} T^{8} \)
19$D_{4}$ \( ( 1 + 13472 T + 134120178 T^{2} + 13472 p^{6} T^{3} + p^{12} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 - 45666180 T^{2} - 26156437245811258 T^{4} - 45666180 p^{12} T^{6} + p^{24} T^{8} \)
29$D_4\times C_2$ \( 1 - 1763439520 T^{2} + 1394728586290536162 T^{4} - 1763439520 p^{12} T^{6} + p^{24} T^{8} \)
31$D_{4}$ \( ( 1 + 28088 T + 538954578 T^{2} + 28088 p^{6} T^{3} + p^{12} T^{4} )^{2} \)
37$D_{4}$ \( ( 1 + 118500 T + 8159603798 T^{2} + 118500 p^{6} T^{3} + p^{12} T^{4} )^{2} \)
41$D_4\times C_2$ \( 1 - 8906980864 T^{2} + 39521221186647918786 T^{4} - 8906980864 p^{12} T^{6} + p^{24} T^{8} \)
43$D_{4}$ \( ( 1 - 1424 T - 9422416878 T^{2} - 1424 p^{6} T^{3} + p^{12} T^{4} )^{2} \)
47$D_4\times C_2$ \( 1 - 25582810180 T^{2} + \)\(32\!\cdots\!02\)\( T^{4} - 25582810180 p^{12} T^{6} + p^{24} T^{8} \)
53$D_4\times C_2$ \( 1 - 20589242016 T^{2} + \)\(92\!\cdots\!46\)\( T^{4} - 20589242016 p^{12} T^{6} + p^{24} T^{8} \)
59$D_4\times C_2$ \( 1 - 122630525220 T^{2} + \)\(71\!\cdots\!42\)\( T^{4} - 122630525220 p^{12} T^{6} + p^{24} T^{8} \)
61$D_{4}$ \( ( 1 - 73412 T + 78569040438 T^{2} - 73412 p^{6} T^{3} + p^{12} T^{4} )^{2} \)
67$D_{4}$ \( ( 1 - 773104 T + 322721854962 T^{2} - 773104 p^{6} T^{3} + p^{12} T^{4} )^{2} \)
71$D_4\times C_2$ \( 1 - 331438972164 T^{2} + \)\(56\!\cdots\!06\)\( T^{4} - 331438972164 p^{12} T^{6} + p^{24} T^{8} \)
73$D_{4}$ \( ( 1 - 140960 T + 281068505058 T^{2} - 140960 p^{6} T^{3} + p^{12} T^{4} )^{2} \)
79$D_{4}$ \( ( 1 - 629240 T + 490074054162 T^{2} - 629240 p^{6} T^{3} + p^{12} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - 1204278186660 T^{2} + \)\(57\!\cdots\!02\)\( T^{4} - 1204278186660 p^{12} T^{6} + p^{24} T^{8} \)
89$D_4\times C_2$ \( 1 - 1438573805824 T^{2} + \)\(93\!\cdots\!86\)\( T^{4} - 1438573805824 p^{12} T^{6} + p^{24} T^{8} \)
97$D_{4}$ \( ( 1 + 661184 T + 1313995880322 T^{2} + 661184 p^{6} T^{3} + p^{12} T^{4} )^{2} \)
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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

−9.469065101840569423293882422599, −9.374600987283723847700072270343, −9.250022768327023919219299327560, −8.445691679005783786012351514839, −8.335313313094586898693054361880, −8.232805648326506072720226247893, −7.82326395834155808051044780065, −7.15307245468968270485602888964, −6.88880699347434111842344994812, −6.67857588228079975407222487890, −6.63211429866077335707665630771, −6.18987727408803657841336297404, −5.41001402871559987587909999003, −5.27094362804046630203484511880, −5.17832418069192971853823606633, −4.43556412831646649653110924514, −4.00816903293575151479528925255, −3.59508293480981589893590821480, −3.59016640170314890797546587957, −2.71832180932509790311614856821, −2.10550101352091554343614675603, −2.10397485713043678910479358031, −1.69779240618242093923938838719, −0.38573563338861791928561612401, −0.02352353707224936580317371565, 0.02352353707224936580317371565, 0.38573563338861791928561612401, 1.69779240618242093923938838719, 2.10397485713043678910479358031, 2.10550101352091554343614675603, 2.71832180932509790311614856821, 3.59016640170314890797546587957, 3.59508293480981589893590821480, 4.00816903293575151479528925255, 4.43556412831646649653110924514, 5.17832418069192971853823606633, 5.27094362804046630203484511880, 5.41001402871559987587909999003, 6.18987727408803657841336297404, 6.63211429866077335707665630771, 6.67857588228079975407222487890, 6.88880699347434111842344994812, 7.15307245468968270485602888964, 7.82326395834155808051044780065, 8.232805648326506072720226247893, 8.335313313094586898693054361880, 8.445691679005783786012351514839, 9.250022768327023919219299327560, 9.374600987283723847700072270343, 9.469065101840569423293882422599

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