Properties

Label 16-300e8-1.1-c1e8-0-3
Degree $16$
Conductor $6.561\times 10^{19}$
Sign $1$
Analytic cond. $1084.39$
Root an. cond. $1.54774$
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-s − 2·9-s + 8·13-s − 3·16-s + 2·36-s − 8·37-s + 36·49-s − 8·52-s − 8·61-s + 3·64-s + 2·81-s + 48·97-s − 40·109-s − 16·117-s − 48·121-s + 127-s + 131-s + 137-s + 139-s + 6·144-s + 8·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + ⋯
L(s)  = 1  − 1/2·4-s − 2/3·9-s + 2.21·13-s − 3/4·16-s + 1/3·36-s − 1.31·37-s + 36/7·49-s − 1.10·52-s − 1.02·61-s + 3/8·64-s + 2/9·81-s + 4.87·97-s − 3.83·109-s − 1.47·117-s − 4.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1/2·144-s + 0.657·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.0760·173-s + ⋯

Functional equation

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

Invariants

Degree: \(16\)
Conductor: \(2^{16} \cdot 3^{8} \cdot 5^{16}\)
Sign: $1$
Analytic conductor: \(1084.39\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{300} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{16} \cdot 3^{8} \cdot 5^{16} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.227255684\)
\(L(\frac12)\) \(\approx\) \(1.227255684\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T^{2} + p^{2} T^{4} + p^{2} T^{6} + p^{4} T^{8} \)
3 \( 1 + 2 T^{2} + 2 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} \)
5 \( 1 \)
good7 \( ( 1 - 18 T^{2} + 162 T^{4} - 18 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
11 \( ( 1 + 24 T^{2} + 318 T^{4} + 24 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
13 \( ( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{4} \)
17 \( ( 1 - 8 T + p T^{2} )^{4}( 1 + 8 T + p T^{2} )^{4} \)
19 \( ( 1 - 56 T^{2} + 1438 T^{4} - 56 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
23 \( ( 1 + 34 T^{2} + 514 T^{4} + 34 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
29 \( ( 1 - 80 T^{2} + 3214 T^{4} - 80 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
31 \( ( 1 - 96 T^{2} + 4158 T^{4} - 96 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
37 \( ( 1 + 2 T + 58 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{4} \)
41 \( ( 1 - 112 T^{2} + 5886 T^{4} - 112 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
43 \( ( 1 - 110 T^{2} + 5890 T^{4} - 110 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
47 \( ( 1 + 178 T^{2} + 12322 T^{4} + 178 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
53 \( ( 1 - 44 T^{2} + 1750 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
59 \( ( 1 - 16 T^{2} + 1518 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
61 \( ( 1 + 2 T + 106 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{4} \)
67 \( ( 1 - 222 T^{2} + 21282 T^{4} - 222 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
71 \( ( 1 + 228 T^{2} + 22806 T^{4} + 228 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
73 \( ( 1 + 78 T^{2} + p^{2} T^{4} )^{4} \)
79 \( ( 1 - 168 T^{2} + 19470 T^{4} - 168 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
83 \( ( 1 + 82 T^{2} + 4834 T^{4} + 82 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
89 \( ( 1 - 212 T^{2} + 25990 T^{4} - 212 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
97 \( ( 1 - 6 T + p T^{2} )^{8} \)
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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

−5.28431288016760838679428698690, −5.12080329401952024716331123853, −5.05193064724565745811613278584, −4.79841076686813089934800056249, −4.66104511109969010822380245253, −4.64225323437421005847002337631, −4.29584253173765608915211451174, −4.13857923000341105035196113773, −3.88413711283470981255955176605, −3.85823842921416065547660384256, −3.78031368100233113485143075041, −3.62676430646168778544878298472, −3.45823220178323003835659713269, −3.35621918157467965496976573047, −2.95329091601968759064014084546, −2.79038990651337344931056864529, −2.52318802995770322804562207815, −2.36764872704517405538761170340, −2.36519375137095092997258297346, −2.17596323524311128721225686552, −1.53736606286210913618572713178, −1.50222307256617399489668037956, −1.07838627622742878055959292716, −1.07712393993001698090376670091, −0.30608622156097079726339614810, 0.30608622156097079726339614810, 1.07712393993001698090376670091, 1.07838627622742878055959292716, 1.50222307256617399489668037956, 1.53736606286210913618572713178, 2.17596323524311128721225686552, 2.36519375137095092997258297346, 2.36764872704517405538761170340, 2.52318802995770322804562207815, 2.79038990651337344931056864529, 2.95329091601968759064014084546, 3.35621918157467965496976573047, 3.45823220178323003835659713269, 3.62676430646168778544878298472, 3.78031368100233113485143075041, 3.85823842921416065547660384256, 3.88413711283470981255955176605, 4.13857923000341105035196113773, 4.29584253173765608915211451174, 4.64225323437421005847002337631, 4.66104511109969010822380245253, 4.79841076686813089934800056249, 5.05193064724565745811613278584, 5.12080329401952024716331123853, 5.28431288016760838679428698690

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

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