Properties

Label 16-12e24-1.1-c2e8-0-0
Degree $16$
Conductor $7.950\times 10^{25}$
Sign $1$
Analytic cond. $2.41562\times 10^{13}$
Root an. cond. $6.86182$
Motivic weight $2$
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  − 88·25-s + 196·49-s − 568·73-s − 200·97-s − 680·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 100·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯
L(s)  = 1  − 3.51·25-s + 4·49-s − 7.78·73-s − 2.06·97-s − 5.61·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 0.591·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + 0.00448·223-s + 0.00440·227-s + 0.00436·229-s + 0.00429·233-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{48} \cdot 3^{24}\)
Sign: $1$
Analytic conductor: \(2.41562\times 10^{13}\)
Root analytic conductor: \(6.86182\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1728} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{48} \cdot 3^{24} ,\ ( \ : [1]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.006141648402\)
\(L(\frac12)\) \(\approx\) \(0.006141648402\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( ( 1 + 22 T^{2} + p^{4} T^{4} )^{4} \)
7 \( ( 1 - p^{2} T^{2} + p^{4} T^{4} )^{4} \)
11 \( ( 1 + 170 T^{2} + p^{4} T^{4} )^{4} \)
13 \( ( 1 + 25 T^{2} + p^{4} T^{4} )^{4} \)
17 \( ( 1 + 362 T^{2} + p^{4} T^{4} )^{4} \)
19 \( ( 1 + 647 T^{2} + p^{4} T^{4} )^{4} \)
23 \( ( 1 - 842 T^{2} + p^{4} T^{4} )^{4} \)
29 \( ( 1 + 910 T^{2} + p^{4} T^{4} )^{4} \)
31 \( ( 1 - 1822 T^{2} + p^{4} T^{4} )^{4} \)
37 \( ( 1 - 2375 T^{2} + p^{4} T^{4} )^{4} \)
41 \( ( 1 - 94 T^{2} + p^{4} T^{4} )^{4} \)
43 \( ( 1 + 1970 T^{2} + p^{4} T^{4} )^{4} \)
47 \( ( 1 + 982 T^{2} + p^{4} T^{4} )^{4} \)
53 \( ( 1 - 5330 T^{2} + p^{4} T^{4} )^{4} \)
59 \( ( 1 + 6314 T^{2} + p^{4} T^{4} )^{4} \)
61 \( ( 1 - 7079 T^{2} + p^{4} T^{4} )^{4} \)
67 \( ( 1 - p^{2} T^{2} + p^{4} T^{4} )^{4} \)
71 \( ( 1 + 3742 T^{2} + p^{4} T^{4} )^{4} \)
73 \( ( 1 + 71 T + p^{2} T^{2} )^{8} \)
79 \( ( 1 + 10319 T^{2} + p^{4} T^{4} )^{4} \)
83 \( ( 1 - 4654 T^{2} + p^{4} T^{4} )^{4} \)
89 \( ( 1 - 10294 T^{2} + p^{4} T^{4} )^{4} \)
97 \( ( 1 + 25 T + p^{2} T^{2} )^{8} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−3.76660575400644437269900305099, −3.59073577326330632765067038176, −3.42933872523375038331995076719, −3.42656809080920182652120076723, −3.41439945873293673447109025019, −2.91432149632903365904808842496, −2.84961421672070501153407091319, −2.82756636959243666788269527181, −2.69928626153238513022453264921, −2.53001806298188319935418489845, −2.47982372484162658418930824924, −2.34646427030698464644893233113, −2.12010177178148880065823325999, −2.08100580542859812053489934032, −1.84714489078494559952366711481, −1.58411404395506051792494799540, −1.49620344808295110670517619386, −1.41692900422705889589498668637, −1.25210031962785412223389780869, −1.13409147605806736278049902632, −1.02290904791605000695517208850, −0.57950369475835166081938675894, −0.44222130663436500459540019270, −0.17894393778318282928850578274, −0.01065779687532857165666677581, 0.01065779687532857165666677581, 0.17894393778318282928850578274, 0.44222130663436500459540019270, 0.57950369475835166081938675894, 1.02290904791605000695517208850, 1.13409147605806736278049902632, 1.25210031962785412223389780869, 1.41692900422705889589498668637, 1.49620344808295110670517619386, 1.58411404395506051792494799540, 1.84714489078494559952366711481, 2.08100580542859812053489934032, 2.12010177178148880065823325999, 2.34646427030698464644893233113, 2.47982372484162658418930824924, 2.53001806298188319935418489845, 2.69928626153238513022453264921, 2.82756636959243666788269527181, 2.84961421672070501153407091319, 2.91432149632903365904808842496, 3.41439945873293673447109025019, 3.42656809080920182652120076723, 3.42933872523375038331995076719, 3.59073577326330632765067038176, 3.76660575400644437269900305099

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

Plot not available for L-functions of degree greater than 10.