Properties

Label 16-180e8-1.1-c2e8-0-2
Degree $16$
Conductor $1.102\times 10^{18}$
Sign $1$
Analytic cond. $334857.$
Root an. cond. $2.21464$
Motivic weight $2$
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  − 5·4-s − 4·5-s + 16-s + 20·20-s + 24·25-s + 184·29-s + 256·41-s − 208·49-s + 304·61-s + 35·64-s − 4·80-s − 560·89-s − 120·100-s − 296·101-s − 608·109-s − 920·116-s + 272·121-s − 204·125-s + 127-s + 131-s + 137-s + 139-s − 736·145-s + 149-s + 151-s + 157-s + 163-s + ⋯
L(s)  = 1  − 5/4·4-s − 4/5·5-s + 1/16·16-s + 20-s + 0.959·25-s + 6.34·29-s + 6.24·41-s − 4.24·49-s + 4.98·61-s + 0.546·64-s − 0.0499·80-s − 6.29·89-s − 6/5·100-s − 2.93·101-s − 5.57·109-s − 7.93·116-s + 2.24·121-s − 1.63·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 5.07·145-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{16} \cdot 3^{16} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(334857.\)
Root analytic conductor: \(2.21464\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{180} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{16} \cdot 3^{16} \cdot 5^{8} ,\ ( \ : [1]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.258893563\)
\(L(\frac12)\) \(\approx\) \(3.258893563\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 5 T^{2} + 3 p^{3} T^{4} + 5 p^{4} T^{6} + p^{8} T^{8} \)
3 \( 1 \)
5 \( ( 1 + 2 T - 6 T^{2} + 2 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
good7 \( ( 1 + 104 T^{2} + 5454 T^{4} + 104 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
11 \( ( 1 - 136 T^{2} + 28206 T^{4} - 136 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
13 \( ( 1 - 592 T^{2} + 144510 T^{4} - 592 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
17 \( ( 1 - 112 T^{2} + 118878 T^{4} - 112 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
19 \( ( 1 - 436 T^{2} + 275334 T^{4} - 436 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
23 \( ( 1 + 76 p T^{2} + 1290726 T^{4} + 76 p^{5} T^{6} + p^{8} T^{8} )^{2} \)
29 \( ( 1 - 46 T + 1698 T^{2} - 46 p^{2} T^{3} + p^{4} T^{4} )^{4} \)
31 \( ( 1 - 1540 T^{2} + 1506054 T^{4} - 1540 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
37 \( ( 1 - 4720 T^{2} + 9299454 T^{4} - 4720 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
41 \( ( 1 - 64 T + 2334 T^{2} - 64 p^{2} T^{3} + p^{4} T^{4} )^{4} \)
43 \( ( 1 + 4868 T^{2} + 12528486 T^{4} + 4868 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
47 \( ( 1 + 5540 T^{2} + 15331014 T^{4} + 5540 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
53 \( ( 1 - 10480 T^{2} + 43220094 T^{4} - 10480 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
59 \( ( 1 + 2744 T^{2} + 25695534 T^{4} + 2744 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
61 \( ( 1 - 38 T + p^{2} T^{2} )^{8} \)
67 \( ( 1 + 8564 T^{2} + 44158854 T^{4} + 8564 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
71 \( ( 1 - 3028 T^{2} - 19410330 T^{4} - 3028 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
73 \( ( 1 - 17140 T^{2} + 129420582 T^{4} - 17140 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
79 \( ( 1 - 22660 T^{2} + 205335174 T^{4} - 22660 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
83 \( ( 1 + 23492 T^{2} + 230783910 T^{4} + 23492 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
89 \( ( 1 + 140 T + 19830 T^{2} + 140 p^{2} T^{3} + p^{4} T^{4} )^{4} \)
97 \( ( 1 + 8252 T^{2} + 163205766 T^{4} + 8252 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
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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

−5.48276417230821024269188304166, −5.35443821781891416711532013046, −5.14189127089276200382354193889, −5.08465120676890535342822308687, −4.73935761620860042045907927933, −4.63701392677076114813841509614, −4.61319451729424821952529536832, −4.18675573081047305299981795492, −4.16880497506865574118289681564, −4.15018697163166027761421982132, −4.04770244209767630608378260453, −3.90022570335255112118159639549, −3.54198900687378576883736996539, −3.04249963774564307185595012233, −2.94040598844136140988719943736, −2.78428631085804593766213294370, −2.72500804699581193060984271347, −2.64438373774194634118169222168, −2.52263218226953585578463943496, −1.71587304703013096225568445561, −1.65352110024393480995398121378, −1.18023235947412593160093871228, −0.885931110136733617412145854802, −0.67878948841024987211227666381, −0.42620846727842713117611088831, 0.42620846727842713117611088831, 0.67878948841024987211227666381, 0.885931110136733617412145854802, 1.18023235947412593160093871228, 1.65352110024393480995398121378, 1.71587304703013096225568445561, 2.52263218226953585578463943496, 2.64438373774194634118169222168, 2.72500804699581193060984271347, 2.78428631085804593766213294370, 2.94040598844136140988719943736, 3.04249963774564307185595012233, 3.54198900687378576883736996539, 3.90022570335255112118159639549, 4.04770244209767630608378260453, 4.15018697163166027761421982132, 4.16880497506865574118289681564, 4.18675573081047305299981795492, 4.61319451729424821952529536832, 4.63701392677076114813841509614, 4.73935761620860042045907927933, 5.08465120676890535342822308687, 5.14189127089276200382354193889, 5.35443821781891416711532013046, 5.48276417230821024269188304166

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

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