Properties

Label 16-42e16-1.1-c3e8-0-2
Degree $16$
Conductor $9.375\times 10^{25}$
Sign $1$
Analytic cond. $1.37694\times 10^{16}$
Root an. cond. $10.2019$
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  + 276·25-s − 312·37-s + 1.18e3·43-s + 1.04e3·67-s − 2.65e3·79-s + 2.60e3·109-s − 780·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.03e4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯
L(s)  = 1  + 2.20·25-s − 1.38·37-s + 4.19·43-s + 1.89·67-s − 3.78·79-s + 2.28·109-s − 0.586·121-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 + 4.70·169-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 + ⋯

Functional equation

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

Invariants

Degree: \(16\)
Conductor: \(2^{16} \cdot 3^{16} \cdot 7^{16}\)
Sign: $1$
Analytic conductor: \(1.37694\times 10^{16}\)
Root analytic conductor: \(10.2019\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1764} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{16} \cdot 3^{16} \cdot 7^{16} ,\ ( \ : [3/2]^{8} ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(54.90911559\)
\(L(\frac12)\) \(\approx\) \(54.90911559\)
\(L(\frac{5}{2})\) 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 \( 1 \)
7 \( 1 \)
good5 \( ( 1 - 138 T^{2} + 3419 T^{4} - 138 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
11 \( ( 1 + 390 T^{2} - 1619461 T^{4} + 390 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
13 \( ( 1 - 2582 T^{2} + p^{6} T^{4} )^{4} \)
17 \( ( 1 - 754 T^{2} - 23569053 T^{4} - 754 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
19 \( ( 1 - 6742 T^{2} - 1591317 T^{4} - 6742 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
23 \( ( 1 + 3134 T^{2} - 138213933 T^{4} + 3134 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
29 \( ( 1 + 36570 T^{2} + p^{6} T^{4} )^{4} \)
31 \( ( 1 - 52606 T^{2} + 1879887555 T^{4} - 52606 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
37 \( ( 1 + 78 T - 44569 T^{2} + 78 p^{3} T^{3} + p^{6} T^{4} )^{4} \)
41 \( ( 1 - 32510 T^{2} + p^{6} T^{4} )^{4} \)
43 \( ( 1 - 148 T + p^{3} T^{2} )^{8} \)
47 \( ( 1 + 9186 T^{2} - 10694832733 T^{4} + 9186 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
53 \( ( 1 - 285546 T^{2} + 59372156987 T^{4} - 285546 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
59 \( ( 1 - 107910 T^{2} - 30535965541 T^{4} - 107910 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
61 \( ( 1 - 112138 T^{2} - 38945443317 T^{4} - 112138 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
67 \( ( 1 - 260 T - 233163 T^{2} - 260 p^{3} T^{3} + p^{6} T^{4} )^{4} \)
71 \( ( 1 + 200034 T^{2} + p^{6} T^{4} )^{4} \)
73 \( ( 1 - 331570 T^{2} - 41395561389 T^{4} - 331570 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
79 \( ( 1 + 664 T - 52143 T^{2} + 664 p^{3} T^{3} + p^{6} T^{4} )^{4} \)
83 \( ( 1 + 1127446 T^{2} + p^{6} T^{4} )^{4} \)
89 \( ( 1 - 638370 T^{2} - 89465034061 T^{4} - 638370 p^{6} T^{6} + p^{12} T^{8} )^{2} \)
97 \( ( 1 + 458050 T^{2} + p^{6} T^{4} )^{4} \)
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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

−3.41055686833476983216716266649, −3.36351095515351381069009291995, −3.15734354982639115983192019874, −3.00918986509797266953335554245, −2.95817950311133866848011260498, −2.95359974689821492923858237243, −2.92259685770942361925310018871, −2.66605454089611740625208235653, −2.59354605374196025154096728909, −2.41417295463118537934013478204, −2.08996894380896648122877870523, −1.94688531055468095574190829706, −1.93161151170237376779680737104, −1.82049762643059120880033544930, −1.76574658729275333721218040177, −1.72473932753797834250155033640, −1.36001185679994170545148623977, −1.08523970694121390581112336426, −0.993131907995984464148247755121, −0.864645094989009620931127662709, −0.72927342379424361094395929349, −0.55234447022579516666689483540, −0.47242750031268482501799104556, −0.43124841097577657604020987384, −0.29425590682355195468331790929, 0.29425590682355195468331790929, 0.43124841097577657604020987384, 0.47242750031268482501799104556, 0.55234447022579516666689483540, 0.72927342379424361094395929349, 0.864645094989009620931127662709, 0.993131907995984464148247755121, 1.08523970694121390581112336426, 1.36001185679994170545148623977, 1.72473932753797834250155033640, 1.76574658729275333721218040177, 1.82049762643059120880033544930, 1.93161151170237376779680737104, 1.94688531055468095574190829706, 2.08996894380896648122877870523, 2.41417295463118537934013478204, 2.59354605374196025154096728909, 2.66605454089611740625208235653, 2.92259685770942361925310018871, 2.95359974689821492923858237243, 2.95817950311133866848011260498, 3.00918986509797266953335554245, 3.15734354982639115983192019874, 3.36351095515351381069009291995, 3.41055686833476983216716266649

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

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