Properties

Label 16-42e16-1.1-c1e8-0-5
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  + 4·2-s + 6·4-s + 4·8-s + 3·16-s + 24·25-s + 32·29-s − 32·37-s + 96·50-s − 16·53-s + 128·58-s − 28·64-s − 128·74-s + 144·100-s − 64·106-s + 192·116-s + 48·121-s + 127-s − 72·128-s + 131-s + 137-s + 139-s − 192·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  + 2.82·2-s + 3·4-s + 1.41·8-s + 3/4·16-s + 24/5·25-s + 5.94·29-s − 5.26·37-s + 13.5·50-s − 2.19·53-s + 16.8·58-s − 7/2·64-s − 14.8·74-s + 72/5·100-s − 6.21·106-s + 17.8·116-s + 4.36·121-s + 0.0887·127-s − 6.36·128-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 15.7·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-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

\(L(1)\) \(\approx\) \(33.88270090\)
\(L(\frac12)\) \(\approx\) \(33.88270090\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 - p T + 3 T^{2} - p^{2} T^{3} + p^{2} T^{4} )^{2} \)
3 \( 1 \)
7 \( 1 \)
good5 \( ( 1 - 12 T^{2} + 78 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
11 \( ( 1 - 24 T^{2} + 354 T^{4} - 24 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
13 \( ( 1 - 44 T^{2} + 814 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
17 \( ( 1 - 48 T^{2} + 1056 T^{4} - 48 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
19 \( ( 1 + 48 T^{2} + 1136 T^{4} + 48 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
23 \( ( 1 - 36 T^{2} + 870 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
29 \( ( 1 - 8 T + 66 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{4} \)
31 \( ( 1 + 28 T^{2} + 1990 T^{4} + 28 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
37 \( ( 1 + 4 T + p T^{2} )^{8} \)
41 \( ( 1 - 128 T^{2} + 7296 T^{4} - 128 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
43 \( ( 1 - 56 T^{2} + 4450 T^{4} - 56 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
47 \( ( 1 + 140 T^{2} + 9286 T^{4} + 140 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
53 \( ( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \)
59 \( ( 1 + 32 T^{2} + 6640 T^{4} + 32 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
61 \( ( 1 - 44 T^{2} + 2926 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
67 \( ( 1 - 156 T^{2} + 13014 T^{4} - 156 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
71 \( ( 1 - p T^{2} )^{8} \)
73 \( ( 1 - 32 T^{2} + 6496 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
79 \( ( 1 - 236 T^{2} + 25894 T^{4} - 236 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
83 \( ( 1 + 224 T^{2} + 25072 T^{4} + 224 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
89 \( ( 1 - 304 T^{2} + 38848 T^{4} - 304 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
97 \( ( 1 - 288 T^{2} + 38304 T^{4} - 288 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
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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

−3.93246217242978761109293089812, −3.83627355501274882655575733580, −3.75485839554832555740473414261, −3.53646425887043738388242517555, −3.48066176705211590264681789042, −3.26748081779968369671768653610, −3.07369086066665307742814317111, −3.01788151796780653131800738279, −3.01304089291078148212495002278, −2.93381265723579301954906721011, −2.85165886021713775276410631828, −2.83113023157235481318387535669, −2.57714951350536271707406359861, −2.34055035396452087219776500703, −1.95363382364094375570057042803, −1.88255628763068660203725018144, −1.84101042599861127917761655037, −1.72041227116108906004761835962, −1.50772180543232620050629272840, −1.46536913053903763277420125619, −0.892584406168074569971257805463, −0.877742154666773418036744179240, −0.816137085359958016218680752719, −0.67469276887324311426059706139, −0.22990299786690244910795053257, 0.22990299786690244910795053257, 0.67469276887324311426059706139, 0.816137085359958016218680752719, 0.877742154666773418036744179240, 0.892584406168074569971257805463, 1.46536913053903763277420125619, 1.50772180543232620050629272840, 1.72041227116108906004761835962, 1.84101042599861127917761655037, 1.88255628763068660203725018144, 1.95363382364094375570057042803, 2.34055035396452087219776500703, 2.57714951350536271707406359861, 2.83113023157235481318387535669, 2.85165886021713775276410631828, 2.93381265723579301954906721011, 3.01304089291078148212495002278, 3.01788151796780653131800738279, 3.07369086066665307742814317111, 3.26748081779968369671768653610, 3.48066176705211590264681789042, 3.53646425887043738388242517555, 3.75485839554832555740473414261, 3.83627355501274882655575733580, 3.93246217242978761109293089812

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

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