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Properties

Label 8-12e8-1.1-c14e4-0-4
Degree $8$
Conductor $429981696$
Sign $1$
Analytic cond. $1.02739\times 10^{9}$
Root an. cond. $13.3803$
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  + 8.70e4·5-s + 6.68e7·13-s − 1.31e7·17-s − 1.68e10·25-s + 1.75e10·29-s − 1.39e11·37-s − 6.62e11·41-s + 1.53e12·49-s + 2.97e12·53-s − 6.62e12·61-s + 5.81e12·65-s − 1.20e13·73-s − 1.14e12·85-s − 1.06e14·89-s + 9.57e13·97-s + 4.60e14·101-s − 7.43e14·109-s − 3.48e14·113-s + 1.33e15·121-s − 2.08e15·125-s + 127-s + 131-s + 137-s + 139-s + 1.52e15·145-s + 149-s + 151-s + ⋯
L(s)  = 1  + 1.11·5-s + 1.06·13-s − 0.0319·17-s − 2.76·25-s + 1.01·29-s − 1.47·37-s − 3.39·41-s + 2.27·49-s + 2.53·53-s − 2.10·61-s + 1.18·65-s − 1.08·73-s − 0.0355·85-s − 2.40·89-s + 1.18·97-s + 4.29·101-s − 4.06·109-s − 1.47·113-s + 3.51·121-s − 4.36·125-s + 1.12·145-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(15-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+7)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(1.02739\times 10^{9}\)
Root analytic conductor: \(13.3803\)
Motivic weight: \(14\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 3^{8} ,\ ( \ : 7, 7, 7, 7 ),\ 1 )\)

Particular Values

\(L(\frac{15}{2})\) \(\approx\) \(0.9082349319\)
\(L(\frac12)\) \(\approx\) \(0.9082349319\)
\(L(8)\) 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 \( 1 \)
good5$D_{4}$ \( ( 1 - 348 p^{3} T + 2254488382 p T^{2} - 348 p^{17} T^{3} + p^{28} T^{4} )^{2} \)
7$D_4\times C_2$ \( 1 - 219949408156 p T^{2} + \)\(25\!\cdots\!42\)\( p^{2} T^{4} - 219949408156 p^{29} T^{6} + p^{56} T^{8} \)
11$D_4\times C_2$ \( 1 - 1334996682152740 T^{2} + \)\(60\!\cdots\!82\)\( p^{2} T^{4} - 1334996682152740 p^{28} T^{6} + p^{56} T^{8} \)
13$D_{4}$ \( ( 1 - 2571716 p T + 2891929558782 p^{3} T^{2} - 2571716 p^{15} T^{3} + p^{28} T^{4} )^{2} \)
17$D_{4}$ \( ( 1 + 6558276 T + 267198985045368902 T^{2} + 6558276 p^{14} T^{3} + p^{28} T^{4} )^{2} \)
19$D_4\times C_2$ \( 1 - 2057925827207927140 T^{2} + \)\(23\!\cdots\!22\)\( T^{4} - 2057925827207927140 p^{28} T^{6} + p^{56} T^{8} \)
23$D_4\times C_2$ \( 1 - 11694673362806264260 T^{2} + \)\(27\!\cdots\!22\)\( T^{4} - 11694673362806264260 p^{28} T^{6} + p^{56} T^{8} \)
29$D_{4}$ \( ( 1 - 8750892876 T + \)\(38\!\cdots\!46\)\( T^{2} - 8750892876 p^{14} T^{3} + p^{28} T^{4} )^{2} \)
31$D_4\times C_2$ \( 1 - \)\(14\!\cdots\!60\)\( T^{2} + \)\(11\!\cdots\!82\)\( T^{4} - \)\(14\!\cdots\!60\)\( p^{28} T^{6} + p^{56} T^{8} \)
37$D_{4}$ \( ( 1 + 69992867180 T + \)\(13\!\cdots\!38\)\( T^{2} + 69992867180 p^{14} T^{3} + p^{28} T^{4} )^{2} \)
41$D_{4}$ \( ( 1 + 331044205284 T + \)\(92\!\cdots\!26\)\( T^{2} + 331044205284 p^{14} T^{3} + p^{28} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - \)\(11\!\cdots\!20\)\( T^{2} + \)\(93\!\cdots\!42\)\( T^{4} - \)\(11\!\cdots\!20\)\( p^{28} T^{6} + p^{56} T^{8} \)
47$D_4\times C_2$ \( 1 - \)\(36\!\cdots\!52\)\( T^{2} + \)\(15\!\cdots\!58\)\( T^{4} - \)\(36\!\cdots\!52\)\( p^{28} T^{6} + p^{56} T^{8} \)
53$D_{4}$ \( ( 1 - 1487586812076 T + \)\(17\!\cdots\!22\)\( T^{2} - 1487586812076 p^{14} T^{3} + p^{28} T^{4} )^{2} \)
59$D_4\times C_2$ \( 1 - \)\(18\!\cdots\!00\)\( T^{2} + \)\(15\!\cdots\!02\)\( T^{4} - \)\(18\!\cdots\!00\)\( p^{28} T^{6} + p^{56} T^{8} \)
61$D_{4}$ \( ( 1 + 3312485425868 T + \)\(20\!\cdots\!98\)\( T^{2} + 3312485425868 p^{14} T^{3} + p^{28} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 - \)\(86\!\cdots\!80\)\( T^{2} + \)\(42\!\cdots\!42\)\( T^{4} - \)\(86\!\cdots\!80\)\( p^{28} T^{6} + p^{56} T^{8} \)
71$D_4\times C_2$ \( 1 - \)\(22\!\cdots\!48\)\( T^{2} + \)\(25\!\cdots\!98\)\( T^{4} - \)\(22\!\cdots\!48\)\( p^{28} T^{6} + p^{56} T^{8} \)
73$D_{4}$ \( ( 1 + 6016566294556 T + \)\(75\!\cdots\!62\)\( T^{2} + 6016566294556 p^{14} T^{3} + p^{28} T^{4} )^{2} \)
79$D_4\times C_2$ \( 1 - \)\(88\!\cdots\!24\)\( T^{2} + \)\(43\!\cdots\!66\)\( T^{4} - \)\(88\!\cdots\!24\)\( p^{28} T^{6} + p^{56} T^{8} \)
83$D_4\times C_2$ \( 1 - \)\(15\!\cdots\!72\)\( T^{2} + \)\(14\!\cdots\!18\)\( T^{4} - \)\(15\!\cdots\!72\)\( p^{28} T^{6} + p^{56} T^{8} \)
89$D_{4}$ \( ( 1 + 53153507248548 T + \)\(31\!\cdots\!18\)\( T^{2} + 53153507248548 p^{14} T^{3} + p^{28} T^{4} )^{2} \)
97$D_{4}$ \( ( 1 - 47896582968068 T + \)\(11\!\cdots\!94\)\( T^{2} - 47896582968068 p^{14} T^{3} + p^{28} T^{4} )^{2} \)
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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

−6.88136818126744869772888708658, −6.73447584099559052809441566407, −6.69402882532787959957886086445, −6.00368260929654481278772506523, −5.88825357857707864001569662159, −5.73995681885321929824739555428, −5.66465914741766585750842013108, −5.22090495810619426563740793419, −4.85403914503935655259012902938, −4.54257050776606243991992001796, −4.42854489073066278664082541238, −3.80300682708623734456322827523, −3.64312301804612530529462276847, −3.52977960996114059754317203881, −3.35188953285948479522716396872, −2.63394833432049356449597326980, −2.46975300364223608787477454059, −2.25834120725428577033017475683, −1.99052529759054769240247681850, −1.61592072281339636769345931144, −1.35727792940403391351816567381, −1.17363306486642321976979825275, −0.972040711968172506312174933535, −0.21575367694890698101065225944, −0.16124038749566923148465644366, 0.16124038749566923148465644366, 0.21575367694890698101065225944, 0.972040711968172506312174933535, 1.17363306486642321976979825275, 1.35727792940403391351816567381, 1.61592072281339636769345931144, 1.99052529759054769240247681850, 2.25834120725428577033017475683, 2.46975300364223608787477454059, 2.63394833432049356449597326980, 3.35188953285948479522716396872, 3.52977960996114059754317203881, 3.64312301804612530529462276847, 3.80300682708623734456322827523, 4.42854489073066278664082541238, 4.54257050776606243991992001796, 4.85403914503935655259012902938, 5.22090495810619426563740793419, 5.66465914741766585750842013108, 5.73995681885321929824739555428, 5.88825357857707864001569662159, 6.00368260929654481278772506523, 6.69402882532787959957886086445, 6.73447584099559052809441566407, 6.88136818126744869772888708658

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