Properties

Label 16-33e16-1.1-c2e8-0-0
Degree $16$
Conductor $1.978\times 10^{24}$
Sign $1$
Analytic cond. $6.01040\times 10^{11}$
Root an. cond. $5.44730$
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  + 16·4-s + 126·16-s + 168·25-s + 96·31-s + 16·37-s − 360·49-s + 720·64-s + 216·67-s + 352·97-s + 2.68e3·100-s + 1.25e3·103-s + 1.53e3·124-s + 127-s + 131-s + 137-s + 139-s + 256·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 332·169-s + 173-s + 179-s + 181-s + 191-s + ⋯
L(s)  = 1  + 4·4-s + 63/8·16-s + 6.71·25-s + 3.09·31-s + 0.432·37-s − 7.34·49-s + 45/4·64-s + 3.22·67-s + 3.62·97-s + 26.8·100-s + 12.1·103-s + 12.3·124-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 1.72·148-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 1.96·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(3^{16} \cdot 11^{16}\)
Sign: $1$
Analytic conductor: \(6.01040\times 10^{11}\)
Root analytic conductor: \(5.44730\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1089} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 3^{16} \cdot 11^{16} ,\ ( \ : [1]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.1084115308\)
\(L(\frac12)\) \(\approx\) \(0.1084115308\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( ( 1 - p^{3} T^{2} + 33 T^{4} - p^{7} T^{6} + p^{8} T^{8} )^{2} \)
5 \( ( 1 - 84 T^{2} + 2999 T^{4} - 84 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
7 \( ( 1 + 180 T^{2} + 12842 T^{4} + 180 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
13 \( ( 1 + 166 T^{2} + 10011 T^{4} + 166 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
17 \( ( 1 - 164 T^{2} + 49551 T^{4} - 164 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
19 \( ( 1 + 1188 T^{2} + 604838 T^{4} + 1188 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
23 \( ( 1 + 240 T^{2} + 150722 T^{4} + 240 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
29 \( ( 1 - 2012 T^{2} + 1998183 T^{4} - 2012 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
31 \( ( 1 - 24 T + 1826 T^{2} - 24 p^{2} T^{3} + p^{4} T^{4} )^{4} \)
37 \( ( 1 - 4 T + 1527 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} )^{4} \)
41 \( ( 1 - 1316 T^{2} + 2228751 T^{4} - 1316 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
43 \( ( 1 + 3300 T^{2} + 7805642 T^{4} + 3300 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
47 \( ( 1 - 6852 T^{2} + 20770838 T^{4} - 6852 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
53 \( ( 1 - 1300 T^{2} + 14880327 T^{4} - 1300 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
59 \( ( 1 - 1720 T^{2} + 18675762 T^{4} - 1720 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
61 \( ( 1 + 10788 T^{2} + 56624678 T^{4} + 10788 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
67 \( ( 1 - 54 T + 3092 T^{2} - 54 p^{2} T^{3} + p^{4} T^{4} )^{4} \)
71 \( ( 1 - 9240 T^{2} + 64218002 T^{4} - 9240 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
73 \( ( 1 - 5820 T^{2} + 62668742 T^{4} - 5820 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
79 \( ( 1 + 4348 T^{2} - 21917562 T^{4} + 4348 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
83 \( ( 1 - 17828 T^{2} + 165802998 T^{4} - 17828 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
89 \( ( 1 - 27468 T^{2} + 309749423 T^{4} - 27468 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
97 \( ( 1 - 88 T + 20019 T^{2} - 88 p^{2} T^{3} + p^{4} T^{4} )^{4} \)
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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

−3.95700598650216172299074139179, −3.57871185876543514126968879247, −3.46544409900859677498883544901, −3.39226580741874765350781342214, −3.33045527701827010289940789300, −3.27047130952678263176487828205, −3.19568332117340919320713653844, −3.06445417037081564482194224257, −3.01124383649700552699108117391, −2.82596984834322918480799851744, −2.43853256572762019495544651512, −2.42441873418209665673020907235, −2.38213585923246679344630475113, −2.11929829071330025289509625771, −2.09201628835237726104402598480, −2.01679304847222135618677704851, −1.98477896599178884780928844724, −1.42658275988193732100407971607, −1.33775754593777926512762051971, −1.16402038894236465820687743056, −1.09465044390140256520129479410, −0.855261749755438570559478729957, −0.76401770791476745414658371733, −0.71287368430350526799616229463, −0.008194257328667644583620167797, 0.008194257328667644583620167797, 0.71287368430350526799616229463, 0.76401770791476745414658371733, 0.855261749755438570559478729957, 1.09465044390140256520129479410, 1.16402038894236465820687743056, 1.33775754593777926512762051971, 1.42658275988193732100407971607, 1.98477896599178884780928844724, 2.01679304847222135618677704851, 2.09201628835237726104402598480, 2.11929829071330025289509625771, 2.38213585923246679344630475113, 2.42441873418209665673020907235, 2.43853256572762019495544651512, 2.82596984834322918480799851744, 3.01124383649700552699108117391, 3.06445417037081564482194224257, 3.19568332117340919320713653844, 3.27047130952678263176487828205, 3.33045527701827010289940789300, 3.39226580741874765350781342214, 3.46544409900859677498883544901, 3.57871185876543514126968879247, 3.95700598650216172299074139179

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

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