Properties

Label 16-1560e8-1.1-c0e8-0-3
Degree $16$
Conductor $3.507\times 10^{25}$
Sign $1$
Analytic cond. $0.134975$
Root an. cond. $0.882349$
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  − 8·43-s − 2·81-s + 8·103-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-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  − 8·43-s − 2·81-s + 8·103-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 5^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 5^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{24} \cdot 3^{8} \cdot 5^{8} \cdot 13^{8}\)
Sign: $1$
Analytic conductor: \(0.134975\)
Root analytic conductor: \(0.882349\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1560} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{24} \cdot 3^{8} \cdot 5^{8} \cdot 13^{8} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8087434306\)
\(L(\frac12)\) \(\approx\) \(0.8087434306\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T^{8} \)
3 \( ( 1 + T^{4} )^{2} \)
5 \( 1 + T^{8} \)
13 \( ( 1 + T^{4} )^{2} \)
good7 \( ( 1 + T^{4} )^{4} \)
11 \( ( 1 + T^{8} )^{2} \)
17 \( ( 1 + T^{4} )^{4} \)
19 \( ( 1 + T^{2} )^{8} \)
23 \( ( 1 + T^{4} )^{4} \)
29 \( ( 1 - T )^{8}( 1 + T )^{8} \)
31 \( ( 1 + T^{2} )^{8} \)
37 \( ( 1 + T^{4} )^{4} \)
41 \( ( 1 + T^{8} )^{2} \)
43 \( ( 1 + T )^{8}( 1 + T^{2} )^{4} \)
47 \( ( 1 + T^{8} )^{2} \)
53 \( ( 1 + T^{4} )^{4} \)
59 \( ( 1 + T^{8} )^{2} \)
61 \( ( 1 + T^{4} )^{4} \)
67 \( ( 1 + T^{4} )^{4} \)
71 \( ( 1 + T^{8} )^{2} \)
73 \( ( 1 + T^{4} )^{4} \)
79 \( ( 1 + T^{4} )^{4} \)
83 \( ( 1 + T^{8} )^{2} \)
89 \( ( 1 + T^{8} )^{2} \)
97 \( ( 1 + T^{4} )^{4} \)
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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

−4.25516670060208125221873554307, −4.14203036302488975250800946887, −3.88045409738657295570573515846, −3.87742855825956738592218482077, −3.69885892486212747686099568515, −3.35543705022065715252031079188, −3.35002569589319896260021749475, −3.33201940402259172931230067895, −3.26046804360944876249085346011, −3.25363107092413021471558226961, −3.13590189295346853960459546997, −2.93842527631062362337481380029, −2.72010490759924103216621579801, −2.36763761140940274544485370535, −2.31629495240296061589942592590, −2.28423962520395933815014765202, −2.06247559250676054708604681768, −1.80732844300405348404991807916, −1.80554722541465229422251812685, −1.78325175309551369784117433868, −1.35983823549704205282968685797, −1.32573228339451575568794014122, −1.21099946057303335924408527341, −0.65481148062080092977819628280, −0.50116809066366794890095799437, 0.50116809066366794890095799437, 0.65481148062080092977819628280, 1.21099946057303335924408527341, 1.32573228339451575568794014122, 1.35983823549704205282968685797, 1.78325175309551369784117433868, 1.80554722541465229422251812685, 1.80732844300405348404991807916, 2.06247559250676054708604681768, 2.28423962520395933815014765202, 2.31629495240296061589942592590, 2.36763761140940274544485370535, 2.72010490759924103216621579801, 2.93842527631062362337481380029, 3.13590189295346853960459546997, 3.25363107092413021471558226961, 3.26046804360944876249085346011, 3.33201940402259172931230067895, 3.35002569589319896260021749475, 3.35543705022065715252031079188, 3.69885892486212747686099568515, 3.87742855825956738592218482077, 3.88045409738657295570573515846, 4.14203036302488975250800946887, 4.25516670060208125221873554307

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

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