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Properties

Label 8-12e8-1.1-c14e4-0-6
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  + 5.52e4·7-s − 1.26e7·13-s − 6.47e7·19-s + 1.13e10·25-s + 2.41e10·31-s − 1.22e11·37-s + 6.94e11·43-s − 2.47e12·49-s − 1.18e13·61-s + 2.02e13·67-s − 2.44e13·73-s + 1.00e13·79-s − 6.98e11·91-s + 2.27e14·97-s − 3.26e14·103-s + 4.94e14·109-s + 8.50e14·121-s + 127-s + 131-s − 3.57e12·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  + 0.0670·7-s − 0.201·13-s − 0.0724·19-s + 1.85·25-s + 0.879·31-s − 1.28·37-s + 2.55·43-s − 3.65·49-s − 3.76·61-s + 3.34·67-s − 2.20·73-s + 0.521·79-s − 0.0135·91-s + 2.81·97-s − 2.65·103-s + 2.70·109-s + 2.23·121-s − 0.00485·133-s − 1.05·169-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

\(L(\frac{15}{2})\) \(\approx\) \(9.979574860\)
\(L(\frac12)\) \(\approx\) \(9.979574860\)
\(L(8)\) 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$D_4\times C_2$ \( 1 - 2264883104 p T^{2} + 4251642902397630834 p^{2} T^{4} - 2264883104 p^{29} T^{6} + p^{56} T^{8} \)
7$D_{4}$ \( ( 1 - 3944 p T + 177002783742 p T^{2} - 3944 p^{15} T^{3} + p^{28} T^{4} )^{2} \)
11$D_4\times C_2$ \( 1 - 850486284336868 T^{2} + \)\(38\!\cdots\!78\)\( p^{2} T^{4} - 850486284336868 p^{28} T^{6} + p^{56} T^{8} \)
13$D_{4}$ \( ( 1 + 486272 p T + 12663911692578 p^{2} T^{2} + 486272 p^{15} T^{3} + p^{28} T^{4} )^{2} \)
17$D_4\times C_2$ \( 1 - 38592333333564800 p T^{2} + \)\(16\!\cdots\!82\)\( T^{4} - 38592333333564800 p^{29} T^{6} + p^{56} T^{8} \)
19$D_{4}$ \( ( 1 + 32377696 T + 1003896409565775666 T^{2} + 32377696 p^{14} T^{3} + p^{28} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 - 44586049724882360452 T^{2} + \)\(76\!\cdots\!18\)\( T^{4} - 44586049724882360452 p^{28} T^{6} + p^{56} T^{8} \)
29$D_4\times C_2$ \( 1 - 36556408674135234080 p T^{2} + \)\(45\!\cdots\!02\)\( T^{4} - 36556408674135234080 p^{29} T^{6} + p^{56} T^{8} \)
31$D_{4}$ \( ( 1 - 12093575096 T + \)\(10\!\cdots\!66\)\( T^{2} - 12093575096 p^{14} T^{3} + p^{28} T^{4} )^{2} \)
37$D_{4}$ \( ( 1 + 61211683748 T + \)\(42\!\cdots\!74\)\( T^{2} + 61211683748 p^{14} T^{3} + p^{28} T^{4} )^{2} \)
41$D_4\times C_2$ \( 1 - \)\(70\!\cdots\!20\)\( T^{2} + \)\(24\!\cdots\!22\)\( T^{4} - \)\(70\!\cdots\!20\)\( p^{28} T^{6} + p^{56} T^{8} \)
43$D_{4}$ \( ( 1 - 347216528240 T + \)\(13\!\cdots\!18\)\( T^{2} - 347216528240 p^{14} T^{3} + p^{28} T^{4} )^{2} \)
47$D_4\times C_2$ \( 1 - \)\(25\!\cdots\!40\)\( T^{2} - \)\(11\!\cdots\!98\)\( T^{4} - \)\(25\!\cdots\!40\)\( p^{28} T^{6} + p^{56} T^{8} \)
53$D_4\times C_2$ \( 1 - \)\(39\!\cdots\!96\)\( T^{2} + \)\(26\!\cdots\!26\)\( T^{4} - \)\(39\!\cdots\!96\)\( p^{28} T^{6} + p^{56} T^{8} \)
59$D_4\times C_2$ \( 1 + \)\(13\!\cdots\!80\)\( T^{2} + \)\(42\!\cdots\!62\)\( T^{4} + \)\(13\!\cdots\!80\)\( p^{28} T^{6} + p^{56} T^{8} \)
61$D_{4}$ \( ( 1 + 5910660774140 T + \)\(23\!\cdots\!62\)\( T^{2} + 5910660774140 p^{14} T^{3} + p^{28} T^{4} )^{2} \)
67$D_{4}$ \( ( 1 - 10148235210512 T + \)\(93\!\cdots\!94\)\( T^{2} - 10148235210512 p^{14} T^{3} + p^{28} T^{4} )^{2} \)
71$D_4\times C_2$ \( 1 - \)\(10\!\cdots\!20\)\( T^{2} + \)\(43\!\cdots\!42\)\( T^{4} - \)\(10\!\cdots\!20\)\( p^{28} T^{6} + p^{56} T^{8} \)
73$D_{4}$ \( ( 1 + 12204077381984 T + \)\(20\!\cdots\!62\)\( T^{2} + 12204077381984 p^{14} T^{3} + p^{28} T^{4} )^{2} \)
79$D_{4}$ \( ( 1 - 5002827136520 T + \)\(44\!\cdots\!42\)\( T^{2} - 5002827136520 p^{14} T^{3} + p^{28} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - \)\(31\!\cdots\!52\)\( T^{2} + \)\(10\!\cdots\!38\)\( T^{4} - \)\(31\!\cdots\!52\)\( p^{28} T^{6} + p^{56} T^{8} \)
89$D_4\times C_2$ \( 1 - \)\(27\!\cdots\!88\)\( T^{2} - \)\(29\!\cdots\!22\)\( T^{4} - \)\(27\!\cdots\!88\)\( p^{28} T^{6} + p^{56} T^{8} \)
97$D_{4}$ \( ( 1 - 113861066168128 T + \)\(14\!\cdots\!54\)\( T^{2} - 113861066168128 p^{14} T^{3} + p^{28} T^{4} )^{2} \)
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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

−7.12471025427402364659195952680, −6.62580534930204227540916164943, −6.49048030462832587497759523504, −6.31458786919058671173662762206, −6.15161659205876295242531266587, −5.52353794811721969111404334167, −5.37710696259587090088490560166, −5.28803969016628585445679678313, −4.71137895133331218140276681700, −4.50685365023545602333774260019, −4.38265960544985714294511818429, −4.28062889378161053877835018097, −3.54988369321358746636169456760, −3.28319120658864620148547211103, −3.13409412895740901060502022235, −3.00745373708155949364297277438, −2.61597488189736373287582169913, −2.23376837517093693736209941570, −1.77948841285639264999294303493, −1.63802666394953853626255730041, −1.61126473080395338211045883934, −0.898456569779899848352788444671, −0.70255911427667230341840896365, −0.48795316715543694838504322420, −0.36763630835164079365298047269, 0.36763630835164079365298047269, 0.48795316715543694838504322420, 0.70255911427667230341840896365, 0.898456569779899848352788444671, 1.61126473080395338211045883934, 1.63802666394953853626255730041, 1.77948841285639264999294303493, 2.23376837517093693736209941570, 2.61597488189736373287582169913, 3.00745373708155949364297277438, 3.13409412895740901060502022235, 3.28319120658864620148547211103, 3.54988369321358746636169456760, 4.28062889378161053877835018097, 4.38265960544985714294511818429, 4.50685365023545602333774260019, 4.71137895133331218140276681700, 5.28803969016628585445679678313, 5.37710696259587090088490560166, 5.52353794811721969111404334167, 6.15161659205876295242531266587, 6.31458786919058671173662762206, 6.49048030462832587497759523504, 6.62580534930204227540916164943, 7.12471025427402364659195952680

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