# Properties

 Label 16-45e16-1.1-c1e8-0-2 Degree $16$ Conductor $2.827\times 10^{26}$ Sign $1$ Analytic cond. $4.67322\times 10^{9}$ Root an. cond. $4.02115$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 4·4-s − 2·11-s + 6·16-s − 4·19-s − 2·29-s − 8·31-s − 10·41-s − 8·44-s + 31·49-s − 34·59-s − 26·61-s − 64-s − 16·71-s − 16·76-s + 14·79-s + 18·89-s + 44·109-s − 8·116-s − 35·121-s − 32·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯
 L(s)  = 1 + 2·4-s − 0.603·11-s + 3/2·16-s − 0.917·19-s − 0.371·29-s − 1.43·31-s − 1.56·41-s − 1.20·44-s + 31/7·49-s − 4.42·59-s − 3.32·61-s − 1/8·64-s − 1.89·71-s − 1.83·76-s + 1.57·79-s + 1.90·89-s + 4.21·109-s − 0.742·116-s − 3.18·121-s − 2.87·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$16$$ Conductor: $$3^{32} \cdot 5^{16}$$ Sign: $1$ Analytic conductor: $$4.67322\times 10^{9}$$ Root analytic conductor: $$4.02115$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(16,\ 3^{32} \cdot 5^{16} ,\ ( \ : [1/2]^{8} ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$12.77681721$$ $$L(\frac12)$$ $$\approx$$ $$12.77681721$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1$$
5 $$1$$
good2 $$1 - p^{2} T^{2} + 5 p T^{4} - 15 T^{6} + 21 T^{8} - 15 p^{2} T^{10} + 5 p^{5} T^{12} - p^{8} T^{14} + p^{8} T^{16}$$
7 $$1 - 31 T^{2} + 496 T^{4} - 5345 T^{6} + 42767 T^{8} - 5345 p^{2} T^{10} + 496 p^{4} T^{12} - 31 p^{6} T^{14} + p^{8} T^{16}$$
11 $$( 1 + T + 19 T^{2} + 74 T^{3} + 167 T^{4} + 74 p T^{5} + 19 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} )^{2}$$
13 $$1 - 40 T^{2} + 874 T^{4} - 13315 T^{6} + 180865 T^{8} - 13315 p^{2} T^{10} + 874 p^{4} T^{12} - 40 p^{6} T^{14} + p^{8} T^{16}$$
17 $$1 - 55 T^{2} + 2044 T^{4} - 50325 T^{6} + 994815 T^{8} - 50325 p^{2} T^{10} + 2044 p^{4} T^{12} - 55 p^{6} T^{14} + p^{8} T^{16}$$
19 $$( 1 + 2 T + 49 T^{2} + 34 T^{3} + 1115 T^{4} + 34 p T^{5} + 49 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2}$$
23 $$1 - 73 T^{2} + 2527 T^{4} - 51820 T^{6} + 985903 T^{8} - 51820 p^{2} T^{10} + 2527 p^{4} T^{12} - 73 p^{6} T^{14} + p^{8} T^{16}$$
29 $$( 1 + T + 76 T^{2} - 55 T^{3} + 2597 T^{4} - 55 p T^{5} + 76 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} )^{2}$$
31 $$( 1 + 4 T + 82 T^{2} + 345 T^{3} + 3405 T^{4} + 345 p T^{5} + 82 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2}$$
37 $$1 - 97 T^{2} + 3667 T^{4} - 33124 T^{6} - 1092101 T^{8} - 33124 p^{2} T^{10} + 3667 p^{4} T^{12} - 97 p^{6} T^{14} + p^{8} T^{16}$$
41 $$( 1 + 5 T + 139 T^{2} + 454 T^{3} + 7829 T^{4} + 454 p T^{5} + 139 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} )^{2}$$
43 $$1 - 148 T^{2} + 13882 T^{4} - 918160 T^{6} + 44775523 T^{8} - 918160 p^{2} T^{10} + 13882 p^{4} T^{12} - 148 p^{6} T^{14} + p^{8} T^{16}$$
47 $$1 - 190 T^{2} + 17287 T^{4} - 1047300 T^{6} + 51910773 T^{8} - 1047300 p^{2} T^{10} + 17287 p^{4} T^{12} - 190 p^{6} T^{14} + p^{8} T^{16}$$
53 $$1 - 196 T^{2} + 19762 T^{4} - 1503447 T^{6} + 91143129 T^{8} - 1503447 p^{2} T^{10} + 19762 p^{4} T^{12} - 196 p^{6} T^{14} + p^{8} T^{16}$$
59 $$( 1 + 17 T + 238 T^{2} + 2095 T^{3} + 18809 T^{4} + 2095 p T^{5} + 238 p^{2} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8} )^{2}$$
61 $$( 1 + 13 T + 241 T^{2} + 2288 T^{3} + 21959 T^{4} + 2288 p T^{5} + 241 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} )^{2}$$
67 $$1 - 319 T^{2} + 50440 T^{4} - 5369705 T^{6} + 418333751 T^{8} - 5369705 p^{2} T^{10} + 50440 p^{4} T^{12} - 319 p^{6} T^{14} + p^{8} T^{16}$$
71 $$( 1 + 8 T + 244 T^{2} + 1441 T^{3} + 24947 T^{4} + 1441 p T^{5} + 244 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2}$$
73 $$1 - 388 T^{2} + 71842 T^{4} - 8513491 T^{6} + 722371129 T^{8} - 8513491 p^{2} T^{10} + 71842 p^{4} T^{12} - 388 p^{6} T^{14} + p^{8} T^{16}$$
79 $$( 1 - 7 T + 283 T^{2} - 1590 T^{3} + 32439 T^{4} - 1590 p T^{5} + 283 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} )^{2}$$
83 $$1 - 340 T^{2} + 59242 T^{4} - 7263475 T^{6} + 683930533 T^{8} - 7263475 p^{2} T^{10} + 59242 p^{4} T^{12} - 340 p^{6} T^{14} + p^{8} T^{16}$$
89 $$( 1 - 9 T + 257 T^{2} - 1998 T^{3} + 31929 T^{4} - 1998 p T^{5} + 257 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} )^{2}$$
97 $$1 - 577 T^{2} + 158407 T^{4} - 27023440 T^{6} + 3138718723 T^{8} - 27023440 p^{2} T^{10} + 158407 p^{4} T^{12} - 577 p^{6} T^{14} + p^{8} T^{16}$$
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$$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

−3.67307300956387166792815452353, −3.61077384732845770049674716242, −3.56803492592950249179983138441, −3.47278880662361088884504973645, −3.42402840280227718299451747433, −3.29680302217456312313001041021, −2.91769773678406536251495639913, −2.74082218441246664874929194333, −2.73110321660343213869172632412, −2.68505374313997770400563664893, −2.63679437891508466686820257125, −2.57528527077166780660866141920, −2.53200072005461911125476286815, −1.98300215446278157705738322319, −1.88614411546273063315691791618, −1.81171299347363335788793195324, −1.79416562186639510609692881839, −1.69143183919361226886144494268, −1.49652722679454975153939088220, −1.46261698546784323059281181563, −1.20091999660649536534909648677, −0.72575795276903801257747343921, −0.50019726831901561547232720397, −0.42769262030207582986946232876, −0.37548616613330096952572053691, 0.37548616613330096952572053691, 0.42769262030207582986946232876, 0.50019726831901561547232720397, 0.72575795276903801257747343921, 1.20091999660649536534909648677, 1.46261698546784323059281181563, 1.49652722679454975153939088220, 1.69143183919361226886144494268, 1.79416562186639510609692881839, 1.81171299347363335788793195324, 1.88614411546273063315691791618, 1.98300215446278157705738322319, 2.53200072005461911125476286815, 2.57528527077166780660866141920, 2.63679437891508466686820257125, 2.68505374313997770400563664893, 2.73110321660343213869172632412, 2.74082218441246664874929194333, 2.91769773678406536251495639913, 3.29680302217456312313001041021, 3.42402840280227718299451747433, 3.47278880662361088884504973645, 3.56803492592950249179983138441, 3.61077384732845770049674716242, 3.67307300956387166792815452353

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

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