Properties

Label 16-12e24-1.1-c3e8-0-1
Degree $16$
Conductor $7.950\times 10^{25}$
Sign $1$
Analytic cond. $1.16755\times 10^{16}$
Root an. cond. $10.0972$
Motivic weight $3$
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  + 376·25-s − 2.15e3·49-s + 2.82e3·73-s + 8.88e3·97-s + 8.77e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.66e4·169-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  + 3.00·25-s − 6.28·49-s + 4.52·73-s + 9.30·97-s + 6.59·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 7.55·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + 0.000361·197-s + 0.000356·199-s + 0.000326·211-s + 0.000300·223-s + 0.000292·227-s + 0.000288·229-s + 0.000281·233-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{48} \cdot 3^{24}\)
Sign: $1$
Analytic conductor: \(1.16755\times 10^{16}\)
Root analytic conductor: \(10.0972\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1728} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{48} \cdot 3^{24} ,\ ( \ : [3/2]^{8} ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(14.92793715\)
\(L(\frac12)\) \(\approx\) \(14.92793715\)
\(L(\frac{5}{2})\) 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 \( ( 1 - 94 T^{2} + p^{6} T^{4} )^{4} \)
7 \( ( 1 + 11 p^{2} T^{2} + p^{6} T^{4} )^{4} \)
11 \( ( 1 - 2194 T^{2} + p^{6} T^{4} )^{4} \)
13 \( ( 1 - 4151 T^{2} + p^{6} T^{4} )^{4} \)
17 \( ( 1 + 5614 T^{2} + p^{6} T^{4} )^{4} \)
19 \( ( 1 - 11317 T^{2} + p^{6} T^{4} )^{4} \)
23 \( ( 1 + 20434 T^{2} + p^{6} T^{4} )^{4} \)
29 \( ( 1 - 48154 T^{2} + p^{6} T^{4} )^{4} \)
31 \( ( 1 + 58994 T^{2} + p^{6} T^{4} )^{4} \)
37 \( ( 1 - 90863 T^{2} + p^{6} T^{4} )^{4} \)
41 \( ( 1 + 18034 T^{2} + p^{6} T^{4} )^{4} \)
43 \( ( 1 - 91414 T^{2} + p^{6} T^{4} )^{4} \)
47 \( ( 1 + 76450 T^{2} + p^{6} T^{4} )^{4} \)
53 \( ( 1 + 32342 T^{2} + p^{6} T^{4} )^{4} \)
59 \( ( 1 - 305458 T^{2} + p^{6} T^{4} )^{4} \)
61 \( ( 1 - 423359 T^{2} + p^{6} T^{4} )^{4} \)
67 \( ( 1 - 543445 T^{2} + p^{6} T^{4} )^{4} \)
71 \( ( 1 + 653422 T^{2} + p^{6} T^{4} )^{4} \)
73 \( ( 1 - 353 T + p^{3} T^{2} )^{8} \)
79 \( ( 1 + 986051 T^{2} + p^{6} T^{4} )^{4} \)
83 \( ( 1 - 65302 T^{2} + p^{6} T^{4} )^{4} \)
89 \( ( 1 + 769246 T^{2} + p^{6} T^{4} )^{4} \)
97 \( ( 1 - 1111 T + p^{3} T^{2} )^{8} \)
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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

−3.58607062302093323537468473073, −3.33816972752876364609978499332, −3.18487018502990446661218229121, −3.13061890625864066466978713679, −3.06908276404574468660651734775, −2.97355334950319396229790873750, −2.97161738130859140239068881109, −2.62246491031474461614997705540, −2.58014942247996528707237590430, −2.27347113409811667818315276527, −2.11440620887095422460155704881, −2.01466011817129021306538712724, −1.96449233058026634101435375619, −1.84732939472268240617542010538, −1.76971503066867916770339443075, −1.67383773945118815475685424529, −1.30164215758602349224721084545, −1.16632989188790747734917891639, −1.13299523070648178100570457943, −0.70752694294225817641841188562, −0.67936230660415850343747577523, −0.59408886306051266162699682845, −0.58893958499190445932714812555, −0.43535291213128157972647480870, −0.13340642119779418075542011671, 0.13340642119779418075542011671, 0.43535291213128157972647480870, 0.58893958499190445932714812555, 0.59408886306051266162699682845, 0.67936230660415850343747577523, 0.70752694294225817641841188562, 1.13299523070648178100570457943, 1.16632989188790747734917891639, 1.30164215758602349224721084545, 1.67383773945118815475685424529, 1.76971503066867916770339443075, 1.84732939472268240617542010538, 1.96449233058026634101435375619, 2.01466011817129021306538712724, 2.11440620887095422460155704881, 2.27347113409811667818315276527, 2.58014942247996528707237590430, 2.62246491031474461614997705540, 2.97161738130859140239068881109, 2.97355334950319396229790873750, 3.06908276404574468660651734775, 3.13061890625864066466978713679, 3.18487018502990446661218229121, 3.33816972752876364609978499332, 3.58607062302093323537468473073

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

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