Properties

Label 16-3200e8-1.1-c1e8-0-1
Degree $16$
Conductor $1.100\times 10^{28}$
Sign $1$
Analytic cond. $1.81725\times 10^{11}$
Root an. cond. $5.05491$
Motivic weight $1$
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  − 4·9-s + 24·41-s + 24·49-s − 26·81-s − 8·89-s + 68·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 56·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯
L(s)  = 1  − 4/3·9-s + 3.74·41-s + 24/7·49-s − 2.88·81-s − 0.847·89-s + 6.18·121-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 + 0.0783·163-s + 0.0773·167-s + 4.30·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{56} \cdot 5^{16}\)
Sign: $1$
Analytic conductor: \(1.81725\times 10^{11}\)
Root analytic conductor: \(5.05491\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3200} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{56} \cdot 5^{16} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(14.09079710\)
\(L(\frac12)\) \(\approx\) \(14.09079710\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( ( 1 + T^{2} + p^{2} T^{4} )^{4} \)
7 \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{4} \)
11 \( ( 1 - 17 T^{2} + p^{2} T^{4} )^{4} \)
13 \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{4} \)
17 \( ( 1 - 9 T^{2} + p^{2} T^{4} )^{4} \)
19 \( ( 1 - 33 T^{2} + p^{2} T^{4} )^{4} \)
23 \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{4} \)
29 \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{4} \)
31 \( ( 1 + p T^{2} )^{8} \)
37 \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{4} \)
41 \( ( 1 - 3 T + p T^{2} )^{8} \)
43 \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{4} \)
47 \( ( 1 - 86 T^{2} + p^{2} T^{4} )^{4} \)
53 \( ( 1 - 54 T^{2} + p^{2} T^{4} )^{4} \)
59 \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{4} \)
61 \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{4} \)
67 \( ( 1 + 9 T^{2} + p^{2} T^{4} )^{4} \)
71 \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{4} \)
73 \( ( 1 + 79 T^{2} + p^{2} T^{4} )^{4} \)
79 \( ( 1 - 42 T^{2} + p^{2} T^{4} )^{4} \)
83 \( ( 1 + 121 T^{2} + p^{2} T^{4} )^{4} \)
89 \( ( 1 + T + p T^{2} )^{8} \)
97 \( ( 1 - 94 T^{2} + p^{2} T^{4} )^{4} \)
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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

−3.59695025774478556589644183123, −3.55773992333957888777873800859, −3.27227736939251968638041781582, −3.07667112476848635233065951633, −3.02810112660903401058232020019, −3.00894460282619592593678535657, −2.86648039038340014433134073054, −2.85842460885069053354522728851, −2.65050762315209851037048858378, −2.58471902532718894014288085888, −2.33440120821207507857513306277, −2.29379610390489729861795386611, −2.16422400914050609095237668200, −2.04834286343848006680247694264, −1.93499989224102457799805911149, −1.64753644880678847841600648491, −1.60966168125923872046140163629, −1.47279546757400289306531069159, −1.17977551937094503185373322435, −1.14720553700608517191676820822, −0.814400254076622232425883088918, −0.70516200040660348069073472688, −0.51024498801620632162235455627, −0.41998611454189192793924161490, −0.36406028847228352297394652487, 0.36406028847228352297394652487, 0.41998611454189192793924161490, 0.51024498801620632162235455627, 0.70516200040660348069073472688, 0.814400254076622232425883088918, 1.14720553700608517191676820822, 1.17977551937094503185373322435, 1.47279546757400289306531069159, 1.60966168125923872046140163629, 1.64753644880678847841600648491, 1.93499989224102457799805911149, 2.04834286343848006680247694264, 2.16422400914050609095237668200, 2.29379610390489729861795386611, 2.33440120821207507857513306277, 2.58471902532718894014288085888, 2.65050762315209851037048858378, 2.85842460885069053354522728851, 2.86648039038340014433134073054, 3.00894460282619592593678535657, 3.02810112660903401058232020019, 3.07667112476848635233065951633, 3.27227736939251968638041781582, 3.55773992333957888777873800859, 3.59695025774478556589644183123

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

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