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Properties

Label 8-48e4-1.1-c16e4-0-1
Degree $8$
Conductor $5308416$
Sign $1$
Analytic cond. $3.68554\times 10^{7}$
Root an. cond. $8.82699$
Motivic weight $16$
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  − 1.40e5·5-s − 2.86e7·9-s − 5.86e8·13-s − 5.71e8·17-s − 3.86e11·25-s − 5.85e11·29-s − 2.77e12·37-s − 9.74e12·41-s + 4.02e12·45-s + 8.88e13·49-s + 9.39e13·53-s + 3.74e14·61-s + 8.23e13·65-s + 1.37e15·73-s + 6.17e14·81-s + 8.01e13·85-s + 2.06e16·89-s + 4.57e16·97-s + 3.93e16·101-s + 5.91e16·109-s + 1.24e17·113-s + 1.68e16·117-s + 1.63e17·121-s + 6.16e16·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  − 0.359·5-s − 2/3·9-s − 0.719·13-s − 0.0818·17-s − 2.53·25-s − 1.17·29-s − 0.788·37-s − 1.22·41-s + 0.239·45-s + 2.67·49-s + 1.50·53-s + 1.95·61-s + 0.258·65-s + 1.71·73-s + 1/3·81-s + 0.0294·85-s + 5.24·89-s + 5.83·97-s + 3.62·101-s + 2.96·109-s + 4.67·113-s + 0.479·117-s + 3.56·121-s + 1.03·125-s + 0.420·145-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(17-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s+8)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(5308416\)    =    \(2^{16} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(3.68554\times 10^{7}\)
Root analytic conductor: \(8.82699\)
Motivic weight: \(16\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 5308416,\ (\ :8, 8, 8, 8),\ 1)\)

Particular Values

\(L(\frac{17}{2})\) \(\approx\) \(2.988839123\)
\(L(\frac12)\) \(\approx\) \(2.988839123\)
\(L(9)\) 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$C_2$ \( ( 1 + p^{15} T^{2} )^{2} \)
good5$D_{4}$ \( ( 1 + 70212 T + 40121922958 p T^{2} + 70212 p^{16} T^{3} + p^{32} T^{4} )^{2} \)
7$D_4\times C_2$ \( 1 - 1813451510116 p^{2} T^{2} + \)\(35\!\cdots\!58\)\( p^{6} T^{4} - 1813451510116 p^{34} T^{6} + p^{64} T^{8} \)
11$D_4\times C_2$ \( 1 - 163743480736580260 T^{2} + \)\(90\!\cdots\!02\)\( p^{2} T^{4} - 163743480736580260 p^{32} T^{6} + p^{64} T^{8} \)
13$D_{4}$ \( ( 1 + 1735756 p^{2} T + 6679213418435574 p^{2} T^{2} + 1735756 p^{18} T^{3} + p^{32} T^{4} )^{2} \)
17$D_{4}$ \( ( 1 + 285502668 T + 55375760720126595494 T^{2} + 285502668 p^{16} T^{3} + p^{32} T^{4} )^{2} \)
19$D_4\times C_2$ \( 1 - \)\(10\!\cdots\!64\)\( T^{2} + \)\(44\!\cdots\!62\)\( T^{4} - \)\(10\!\cdots\!64\)\( p^{32} T^{6} + p^{64} T^{8} \)
23$D_4\times C_2$ \( 1 - \)\(14\!\cdots\!52\)\( T^{2} + \)\(12\!\cdots\!02\)\( T^{4} - \)\(14\!\cdots\!52\)\( p^{32} T^{6} + p^{64} T^{8} \)
29$D_{4}$ \( ( 1 + 292810708212 T + \)\(51\!\cdots\!22\)\( T^{2} + 292810708212 p^{16} T^{3} + p^{32} T^{4} )^{2} \)
31$D_4\times C_2$ \( 1 - \)\(17\!\cdots\!80\)\( T^{2} + \)\(14\!\cdots\!22\)\( T^{4} - \)\(17\!\cdots\!80\)\( p^{32} T^{6} + p^{64} T^{8} \)
37$D_{4}$ \( ( 1 + 1385235255548 T + \)\(14\!\cdots\!22\)\( T^{2} + 1385235255548 p^{16} T^{3} + p^{32} T^{4} )^{2} \)
41$D_{4}$ \( ( 1 + 4872348556812 T + \)\(62\!\cdots\!94\)\( T^{2} + 4872348556812 p^{16} T^{3} + p^{32} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - \)\(31\!\cdots\!04\)\( T^{2} + \)\(61\!\cdots\!22\)\( T^{4} - \)\(31\!\cdots\!04\)\( p^{32} T^{6} + p^{64} T^{8} \)
47$D_4\times C_2$ \( 1 - \)\(21\!\cdots\!64\)\( T^{2} + \)\(17\!\cdots\!02\)\( T^{4} - \)\(21\!\cdots\!64\)\( p^{32} T^{6} + p^{64} T^{8} \)
53$D_{4}$ \( ( 1 - 46975486697100 T + \)\(77\!\cdots\!86\)\( T^{2} - 46975486697100 p^{16} T^{3} + p^{32} T^{4} )^{2} \)
59$D_4\times C_2$ \( 1 - \)\(78\!\cdots\!60\)\( T^{2} + \)\(24\!\cdots\!62\)\( T^{4} - \)\(78\!\cdots\!60\)\( p^{32} T^{6} + p^{64} T^{8} \)
61$D_{4}$ \( ( 1 - 187081173156196 T + \)\(22\!\cdots\!02\)\( T^{2} - 187081173156196 p^{16} T^{3} + p^{32} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 + \)\(18\!\cdots\!96\)\( T^{2} + \)\(45\!\cdots\!22\)\( T^{4} + \)\(18\!\cdots\!96\)\( p^{32} T^{6} + p^{64} T^{8} \)
71$D_4\times C_2$ \( 1 - \)\(50\!\cdots\!84\)\( T^{2} + \)\(75\!\cdots\!22\)\( T^{4} - \)\(50\!\cdots\!84\)\( p^{32} T^{6} + p^{64} T^{8} \)
73$D_{4}$ \( ( 1 - 689974858272260 T + \)\(10\!\cdots\!86\)\( T^{2} - 689974858272260 p^{16} T^{3} + p^{32} T^{4} )^{2} \)
79$D_4\times C_2$ \( 1 - \)\(58\!\cdots\!16\)\( T^{2} + \)\(18\!\cdots\!90\)\( T^{4} - \)\(58\!\cdots\!16\)\( p^{32} T^{6} + p^{64} T^{8} \)
83$D_4\times C_2$ \( 1 - \)\(14\!\cdots\!68\)\( T^{2} + \)\(94\!\cdots\!78\)\( T^{4} - \)\(14\!\cdots\!68\)\( p^{32} T^{6} + p^{64} T^{8} \)
89$D_{4}$ \( ( 1 - 10333279314316932 T + \)\(52\!\cdots\!78\)\( T^{2} - 10333279314316932 p^{16} T^{3} + p^{32} T^{4} )^{2} \)
97$D_{4}$ \( ( 1 - 22856009041316420 T + \)\(25\!\cdots\!58\)\( T^{2} - 22856009041316420 p^{16} T^{3} + p^{32} T^{4} )^{2} \)
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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

−8.618240043885811050711211454956, −7.64952462279412294150800208921, −7.58434566357667481498786200258, −7.45670223527929715201828235175, −7.37004042987521028905699257540, −6.74651464549200882498748123583, −6.12924831322160617795791312833, −6.02089859849640008218403654781, −5.97237937388445877514528117082, −5.45349027786787418213711074836, −4.85973798952524210455834229786, −4.77889434722127173524023625977, −4.71052432045709622590226202616, −3.73784114846193849339579340085, −3.54432838761156332739151994531, −3.53670297657709487747972958832, −3.44433223266133500126060916990, −2.23773805320163551158854854031, −2.20757922059709369257527719095, −2.12963540060450191706320775637, −2.05417949892408489313833385916, −1.07053310363622544697666395004, −0.69635231894844950237929421598, −0.61548304338691277365798794316, −0.25347216797906775683724147926, 0.25347216797906775683724147926, 0.61548304338691277365798794316, 0.69635231894844950237929421598, 1.07053310363622544697666395004, 2.05417949892408489313833385916, 2.12963540060450191706320775637, 2.20757922059709369257527719095, 2.23773805320163551158854854031, 3.44433223266133500126060916990, 3.53670297657709487747972958832, 3.54432838761156332739151994531, 3.73784114846193849339579340085, 4.71052432045709622590226202616, 4.77889434722127173524023625977, 4.85973798952524210455834229786, 5.45349027786787418213711074836, 5.97237937388445877514528117082, 6.02089859849640008218403654781, 6.12924831322160617795791312833, 6.74651464549200882498748123583, 7.37004042987521028905699257540, 7.45670223527929715201828235175, 7.58434566357667481498786200258, 7.64952462279412294150800208921, 8.618240043885811050711211454956

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