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Properties

Label 16-539e8-1.1-c0e8-0-0
Degree $16$
Conductor $7.124\times 10^{21}$
Sign $1$
Analytic cond. $2.74136\times 10^{-5}$
Root an. cond. $0.518648$
Motivic weight $0$
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 − 9-s + 11-s + 16-s − 2·23-s − 25-s − 36-s − 3·37-s + 44-s + 3·53-s − 2·67-s − 6·71-s + 5·79-s + 81-s − 2·92-s − 99-s − 100-s + 5·107-s − 4·113-s + 121-s + 127-s + 131-s + 137-s + 139-s − 144-s − 3·148-s + 149-s + ⋯
L(s)  = 1  + 4-s − 9-s + 11-s + 16-s − 2·23-s − 25-s − 36-s − 3·37-s + 44-s + 3·53-s − 2·67-s − 6·71-s + 5·79-s + 81-s − 2·92-s − 99-s − 100-s + 5·107-s − 4·113-s + 121-s + 127-s + 131-s + 137-s + 139-s − 144-s − 3·148-s + 149-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{16} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{16} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(7^{16} \cdot 11^{8}\)
Sign: $1$
Analytic conductor: \(2.74136\times 10^{-5}\)
Root analytic conductor: \(0.518648\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 7^{16} \cdot 11^{8} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3904176653\)
\(L(\frac12)\) \(\approx\) \(0.3904176653\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} \)
good2 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
3 \( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \)
5 \( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \)
13 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
17 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
19 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
23 \( ( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2} \)
29 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
31 \( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \)
37 \( ( 1 + T + T^{2} )^{4}( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} ) \)
41 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
43 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
47 \( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \)
53 \( ( 1 - T + T^{2} )^{4}( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
59 \( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \)
61 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
67 \( ( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2} \)
71 \( ( 1 + T )^{8}( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
73 \( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} ) \)
79 \( ( 1 - T + T^{2} )^{4}( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} ) \)
83 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
89 \( ( 1 - T^{2} + T^{4} )^{4} \)
97 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
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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

−4.98595941483546555414676837602, −4.81247006241702557322958480108, −4.77470590017601454740527618068, −4.68010732806753859639343612687, −4.66114330044270838155498441810, −4.24364941433408646529921838562, −3.98261436155965862808452348239, −3.81843166174025678983195177554, −3.80569367519457982868022962636, −3.75631622455988315815482689394, −3.69888058083738671154768555179, −3.54274479903504429370919957088, −3.39904340069876059459802204054, −2.97448692667774850698703238355, −2.82401808763291648007453776322, −2.75273007634118709112347132324, −2.71942748545024265487422432003, −2.52301066537474133895342193263, −2.11354608418545423259098655529, −2.07756325611349713499780136593, −1.83490627517020706768663209876, −1.64905195630532839796335766350, −1.61847846942603438357443335395, −1.26451736274110363133510564526, −0.904180718382114109564293680756, 0.904180718382114109564293680756, 1.26451736274110363133510564526, 1.61847846942603438357443335395, 1.64905195630532839796335766350, 1.83490627517020706768663209876, 2.07756325611349713499780136593, 2.11354608418545423259098655529, 2.52301066537474133895342193263, 2.71942748545024265487422432003, 2.75273007634118709112347132324, 2.82401808763291648007453776322, 2.97448692667774850698703238355, 3.39904340069876059459802204054, 3.54274479903504429370919957088, 3.69888058083738671154768555179, 3.75631622455988315815482689394, 3.80569367519457982868022962636, 3.81843166174025678983195177554, 3.98261436155965862808452348239, 4.24364941433408646529921838562, 4.66114330044270838155498441810, 4.68010732806753859639343612687, 4.77470590017601454740527618068, 4.81247006241702557322958480108, 4.98595941483546555414676837602

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

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