Properties

Label 16-42e16-1.1-c1e8-0-2
Degree $16$
Conductor $9.375\times 10^{25}$
Sign $1$
Analytic cond. $1.54954\times 10^{9}$
Root an. cond. $3.75308$
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  − 16-s + 40·25-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 104·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + 251-s + ⋯
L(s)  = 1  − 1/4·16-s + 8·25-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 − 8·169-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 + 0.0655·233-s + 0.0646·239-s + 0.0644·241-s + 0.0631·251-s + ⋯

Functional equation

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

Invariants

Degree: \(16\)
Conductor: \(2^{16} \cdot 3^{16} \cdot 7^{16}\)
Sign: $1$
Analytic conductor: \(1.54954\times 10^{9}\)
Root analytic conductor: \(3.75308\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1764} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{16} \cdot 3^{16} \cdot 7^{16} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.409517167\)
\(L(\frac12)\) \(\approx\) \(2.409517167\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T^{4} + p^{4} T^{8} \)
3 \( 1 \)
7 \( 1 \)
good5 \( ( 1 - p T^{2} )^{8} \)
11 \( ( 1 - 206 T^{4} + p^{4} T^{8} )^{2} \)
13 \( ( 1 + p T^{2} )^{8} \)
17 \( ( 1 - p T^{2} )^{8} \)
19 \( ( 1 - p T^{2} )^{8} \)
23 \( ( 1 - 734 T^{4} + p^{4} T^{8} )^{2} \)
29 \( ( 1 + 1234 T^{4} + p^{4} T^{8} )^{2} \)
31 \( ( 1 - p T^{2} )^{8} \)
37 \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{4} \)
41 \( ( 1 - p T^{2} )^{8} \)
43 \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{4} \)
47 \( ( 1 + p T^{2} )^{8} \)
53 \( ( 1 - 5582 T^{4} + p^{4} T^{8} )^{2} \)
59 \( ( 1 + p T^{2} )^{8} \)
61 \( ( 1 + p T^{2} )^{8} \)
67 \( ( 1 - 4 T + p T^{2} )^{4}( 1 + 4 T + p T^{2} )^{4} \)
71 \( ( 1 + 2914 T^{4} + p^{4} T^{8} )^{2} \)
73 \( ( 1 + p T^{2} )^{8} \)
79 \( ( 1 - 8 T + p T^{2} )^{4}( 1 + 8 T + p T^{2} )^{4} \)
83 \( ( 1 + p T^{2} )^{8} \)
89 \( ( 1 - p T^{2} )^{8} \)
97 \( ( 1 + p T^{2} )^{8} \)
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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

−3.91994937968984945359937857199, −3.76811134878409403669927393250, −3.65144859355296760302242581104, −3.51956736401028740466167290689, −3.40918517280381078706709738677, −3.20573116126984107624550954939, −3.11399202297402287835073104776, −2.92614850017699024705418321777, −2.91191697271082113680917885927, −2.90850486927284505701811177868, −2.63173646203382058077457704177, −2.53111625400919786979314569549, −2.48131658203164216809990923619, −2.24579351384033417156779402665, −2.18229772609466143605242245771, −1.94264850714508135166459931508, −1.63726291331363011581626885102, −1.48428093314435314218597358849, −1.45328027664496390013844445215, −1.20723827344751300427739586131, −0.953459356125083143976812203477, −0.914093875373577056563662573168, −0.867907152865859272279176720612, −0.54024605251455427120994492065, −0.11909148935263438302611734128, 0.11909148935263438302611734128, 0.54024605251455427120994492065, 0.867907152865859272279176720612, 0.914093875373577056563662573168, 0.953459356125083143976812203477, 1.20723827344751300427739586131, 1.45328027664496390013844445215, 1.48428093314435314218597358849, 1.63726291331363011581626885102, 1.94264850714508135166459931508, 2.18229772609466143605242245771, 2.24579351384033417156779402665, 2.48131658203164216809990923619, 2.53111625400919786979314569549, 2.63173646203382058077457704177, 2.90850486927284505701811177868, 2.91191697271082113680917885927, 2.92614850017699024705418321777, 3.11399202297402287835073104776, 3.20573116126984107624550954939, 3.40918517280381078706709738677, 3.51956736401028740466167290689, 3.65144859355296760302242581104, 3.76811134878409403669927393250, 3.91994937968984945359937857199

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

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