Properties

Label 16-1045e8-1.1-c0e8-0-1
Degree $16$
Conductor $1.422\times 10^{24}$
Sign $1$
Analytic cond. $0.00547251$
Root an. cond. $0.722165$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s − 2·7-s + 16-s − 8·17-s + 2·19-s + 2·23-s + 25-s + 4·35-s − 2·43-s − 2·47-s − 3·49-s − 2·73-s − 2·80-s + 81-s + 2·83-s + 16·85-s − 4·95-s − 2·112-s − 4·115-s + 16·119-s + 121-s + 127-s + 131-s − 4·133-s + 137-s + 139-s + 149-s + ⋯
L(s)  = 1  − 2·5-s − 2·7-s + 16-s − 8·17-s + 2·19-s + 2·23-s + 25-s + 4·35-s − 2·43-s − 2·47-s − 3·49-s − 2·73-s − 2·80-s + 81-s + 2·83-s + 16·85-s − 4·95-s − 2·112-s − 4·115-s + 16·119-s + 121-s + 127-s + 131-s − 4·133-s + 137-s + 139-s + 149-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 11^{8} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 11^{8} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(5^{8} \cdot 11^{8} \cdot 19^{8}\)
Sign: $1$
Analytic conductor: \(0.00547251\)
Root analytic conductor: \(0.722165\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1045} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 5^{8} \cdot 11^{8} \cdot 19^{8} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.08821310472\)
\(L(\frac12)\) \(\approx\) \(0.08821310472\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
11 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
19 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
good2 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \)
3 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \)
7 \( ( 1 + T^{2} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
13 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \)
17 \( ( 1 + T )^{8}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
23 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
29 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
31 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
37 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \)
41 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
43 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
47 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
53 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \)
59 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
61 \( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
67 \( ( 1 + T^{4} )^{4} \)
71 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
73 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
79 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
83 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \)
89 \( ( 1 + T^{2} )^{8} \)
97 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \)
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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

−4.41031743435027963341026339995, −4.36989187883394538345081472708, −4.21794761450044827675811404365, −4.16719951670227916784774001988, −3.89595447698094866251840859137, −3.87805617599714412630245613422, −3.70503937752948131578165567643, −3.56061288568117692154673103124, −3.52603094134415781395965133426, −3.19737034919313375151934384878, −3.16636252403556442422088020047, −3.15983365218579098004186626335, −2.82967988266207884503276162076, −2.81605443740388489398273946657, −2.75249697126039426628457124516, −2.45028556393840762608289219288, −2.36753192514365595239337081756, −2.09198193242867874593286432338, −1.88880212117456364992933003052, −1.81024450389923655527135178909, −1.64033749352324593505007525334, −1.32945827878924088114668928209, −1.30790359288508258780122014736, −0.47678583538488655542035662149, −0.36420530905654005016788826966, 0.36420530905654005016788826966, 0.47678583538488655542035662149, 1.30790359288508258780122014736, 1.32945827878924088114668928209, 1.64033749352324593505007525334, 1.81024450389923655527135178909, 1.88880212117456364992933003052, 2.09198193242867874593286432338, 2.36753192514365595239337081756, 2.45028556393840762608289219288, 2.75249697126039426628457124516, 2.81605443740388489398273946657, 2.82967988266207884503276162076, 3.15983365218579098004186626335, 3.16636252403556442422088020047, 3.19737034919313375151934384878, 3.52603094134415781395965133426, 3.56061288568117692154673103124, 3.70503937752948131578165567643, 3.87805617599714412630245613422, 3.89595447698094866251840859137, 4.16719951670227916784774001988, 4.21794761450044827675811404365, 4.36989187883394538345081472708, 4.41031743435027963341026339995

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

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