Properties

Label 16-1620e8-1.1-c2e8-0-1
Degree $16$
Conductor $4.744\times 10^{25}$
Sign $1$
Analytic cond. $1.44145\times 10^{13}$
Root an. cond. $6.64392$
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  + 3·5-s − 12·17-s + 12·19-s − 30·23-s − 16·31-s + 48·47-s − 137·49-s + 384·53-s − 38·61-s + 6·79-s − 288·83-s − 36·85-s + 36·95-s − 36·107-s − 452·109-s − 564·113-s − 90·115-s − 129·121-s − 177·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 48·155-s + 157-s + ⋯
L(s)  = 1  + 3/5·5-s − 0.705·17-s + 0.631·19-s − 1.30·23-s − 0.516·31-s + 1.02·47-s − 2.79·49-s + 7.24·53-s − 0.622·61-s + 6/79·79-s − 3.46·83-s − 0.423·85-s + 0.378·95-s − 0.336·107-s − 4.14·109-s − 4.99·113-s − 0.782·115-s − 1.06·121-s − 1.41·125-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 − 0.309·155-s + 0.00636·157-s + ⋯

Functional equation

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

Invariants

Degree: \(16\)
Conductor: \(2^{16} \cdot 3^{32} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(1.44145\times 10^{13}\)
Root analytic conductor: \(6.64392\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1620} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{16} \cdot 3^{32} \cdot 5^{8} ,\ ( \ : [1]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.09743664129\)
\(L(\frac12)\) \(\approx\) \(0.09743664129\)
\(L(2)\) 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 \( 1 \)
5 \( 1 - 3 T + 9 T^{2} + 6 p^{2} T^{3} - 34 p^{2} T^{4} + 6 p^{4} T^{5} + 9 p^{4} T^{6} - 3 p^{6} T^{7} + p^{8} T^{8} \)
good7 \( 1 + 137 T^{2} + 9745 T^{4} + 578414 T^{6} + 30603406 T^{8} + 578414 p^{4} T^{10} + 9745 p^{8} T^{12} + 137 p^{12} T^{14} + p^{16} T^{16} \)
11 \( 1 + 129 T^{2} + 6241 T^{4} - 2435778 T^{6} - 349839762 T^{8} - 2435778 p^{4} T^{10} + 6241 p^{8} T^{12} + 129 p^{12} T^{14} + p^{16} T^{16} \)
13 \( 1 + 136 T^{2} + 1882 p T^{4} - 8580512 T^{6} - 1308354077 T^{8} - 8580512 p^{4} T^{10} + 1882 p^{9} T^{12} + 136 p^{12} T^{14} + p^{16} T^{16} \)
17 \( ( 1 + 3 T + 528 T^{2} + 3 p^{2} T^{3} + p^{4} T^{4} )^{4} \)
19 \( ( 1 - 3 T + 254 T^{2} - 3 p^{2} T^{3} + p^{4} T^{4} )^{4} \)
23 \( ( 1 + 15 T - 837 T^{2} + 60 T^{3} + 728978 T^{4} + 60 p^{2} T^{5} - 837 p^{4} T^{6} + 15 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
29 \( ( 1 + 98 T^{2} - 697677 T^{4} + 98 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
31 \( ( 1 + 8 T + 7 T^{2} - 14920 T^{3} - 981776 T^{4} - 14920 p^{2} T^{5} + 7 p^{4} T^{6} + 8 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
37 \( ( 1 - 1522 T^{2} + p^{4} T^{4} )^{4} \)
41 \( ( 1 + 3186 T^{2} + 7324835 T^{4} + 3186 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
43 \( 1 + 776 T^{2} - 5114414 T^{4} - 869905312 T^{6} + 18932481039523 T^{8} - 869905312 p^{4} T^{10} - 5114414 p^{8} T^{12} + 776 p^{12} T^{14} + p^{16} T^{16} \)
47 \( ( 1 - 24 T - 3150 T^{2} + 16608 T^{3} + 7731011 T^{4} + 16608 p^{2} T^{5} - 3150 p^{4} T^{6} - 24 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
53 \( ( 1 - 96 T + 6041 T^{2} - 96 p^{2} T^{3} + p^{4} T^{4} )^{4} \)
59 \( 1 + 7044 T^{2} + 20683306 T^{4} + 33106151952 T^{6} + 89143933045683 T^{8} + 33106151952 p^{4} T^{10} + 20683306 p^{8} T^{12} + 7044 p^{12} T^{14} + p^{16} T^{16} \)
61 \( ( 1 + 19 T - 6701 T^{2} - 7220 T^{3} + 34682722 T^{4} - 7220 p^{2} T^{5} - 6701 p^{4} T^{6} + 19 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
67 \( 1 + 6584 T^{2} + 23998450 T^{4} - 137945571424 T^{6} - 905153797681181 T^{8} - 137945571424 p^{4} T^{10} + 23998450 p^{8} T^{12} + 6584 p^{12} T^{14} + p^{16} T^{16} \)
71 \( ( 1 - 3208 T^{2} + 50078094 T^{4} - 3208 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
73 \( ( 1 - 9521 T^{2} + 50769360 T^{4} - 9521 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
79 \( ( 1 - 3 T - 8243 T^{2} + 12690 T^{3} + 29089254 T^{4} + 12690 p^{2} T^{5} - 8243 p^{4} T^{6} - 3 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
83 \( ( 1 + 144 T + 6999 T^{2} - 5904 T^{3} - 1603456 T^{4} - 5904 p^{2} T^{5} + 6999 p^{4} T^{6} + 144 p^{6} T^{7} + p^{8} T^{8} )^{2} \)
89 \( ( 1 - 6984 T^{2} + 4586510 T^{4} - 6984 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
97 \( 1 - 1879 T^{2} - 50655359 T^{4} + 230877543998 T^{6} - 5213876107576082 T^{8} + 230877543998 p^{4} T^{10} - 50655359 p^{8} T^{12} - 1879 p^{12} T^{14} + p^{16} T^{16} \)
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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

−3.86629491812680110806226787298, −3.71427235244043478147694328730, −3.53862448186741028213085489403, −3.30693388699230207718030736038, −3.30304529723378554087467008721, −3.11635675199010142747503545683, −2.92294595704197794348286656476, −2.74219356938911570417820462814, −2.72353614691780506157304655737, −2.51309077277415345768404518314, −2.43764850633398577759564245037, −2.31053761442247988724982982140, −2.29274716801329512256576936074, −2.01265397277847014512684585265, −1.99620602658490960523931187767, −1.66809672148179132779352673041, −1.58610014275743835822756749359, −1.21178155839639261663075071651, −1.17103388516827104620984622385, −1.14658355645684011104024560878, −1.12589135929034622325486357764, −0.823548246831678816500947861841, −0.39530192224017506427680425247, −0.12547124811894676714906590408, −0.05509878630055200340151081481, 0.05509878630055200340151081481, 0.12547124811894676714906590408, 0.39530192224017506427680425247, 0.823548246831678816500947861841, 1.12589135929034622325486357764, 1.14658355645684011104024560878, 1.17103388516827104620984622385, 1.21178155839639261663075071651, 1.58610014275743835822756749359, 1.66809672148179132779352673041, 1.99620602658490960523931187767, 2.01265397277847014512684585265, 2.29274716801329512256576936074, 2.31053761442247988724982982140, 2.43764850633398577759564245037, 2.51309077277415345768404518314, 2.72353614691780506157304655737, 2.74219356938911570417820462814, 2.92294595704197794348286656476, 3.11635675199010142747503545683, 3.30304529723378554087467008721, 3.30693388699230207718030736038, 3.53862448186741028213085489403, 3.71427235244043478147694328730, 3.86629491812680110806226787298

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

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