Properties

Label 16-363e8-1.1-c1e8-0-4
Degree $16$
Conductor $3.015\times 10^{20}$
Sign $1$
Analytic cond. $4982.75$
Root an. cond. $1.70251$
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  − 3-s + 4·4-s + 3·9-s − 4·12-s + 4·16-s + 25-s − 10·31-s + 12·36-s + 14·37-s − 4·48-s − 14·49-s − 104·67-s − 75-s + 10·93-s − 34·97-s + 4·100-s + 8·103-s − 14·111-s − 40·124-s + 127-s + 131-s + 137-s + 139-s + 12·144-s + 14·147-s + 56·148-s + 149-s + ⋯
L(s)  = 1  − 0.577·3-s + 2·4-s + 9-s − 1.15·12-s + 16-s + 1/5·25-s − 1.79·31-s + 2·36-s + 2.30·37-s − 0.577·48-s − 2·49-s − 12.7·67-s − 0.115·75-s + 1.03·93-s − 3.45·97-s + 2/5·100-s + 0.788·103-s − 1.32·111-s − 3.59·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 144-s + 1.15·147-s + 4.60·148-s + 0.0819·149-s + ⋯

Functional equation

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

Invariants

Degree: \(16\)
Conductor: \(3^{8} \cdot 11^{16}\)
Sign: $1$
Analytic conductor: \(4982.75\)
Root analytic conductor: \(1.70251\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{363} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 3^{8} \cdot 11^{16} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.147181805\)
\(L(\frac12)\) \(\approx\) \(2.147181805\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T - 2 T^{2} - 5 T^{3} + T^{4} - 5 p T^{5} - 2 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
11 \( 1 \)
good2 \( ( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} )^{2} \)
5 \( ( 1 - 3 T + 4 T^{2} + 3 T^{3} - 29 T^{4} + 3 p T^{5} + 4 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} )( 1 + 3 T + 4 T^{2} - 3 T^{3} - 29 T^{4} - 3 p T^{5} + 4 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} ) \)
7 \( ( 1 + p T^{2} + p^{2} T^{4} + p^{3} T^{6} + p^{4} T^{8} )^{2} \)
13 \( ( 1 + p T^{2} + p^{2} T^{4} + p^{3} T^{6} + p^{4} T^{8} )^{2} \)
17 \( ( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} )^{2} \)
19 \( ( 1 + p T^{2} + p^{2} T^{4} + p^{3} T^{6} + p^{4} T^{8} )^{2} \)
23 \( ( 1 - 9 T + p T^{2} )^{4}( 1 + 9 T + p T^{2} )^{4} \)
29 \( ( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} )^{2} \)
31 \( ( 1 + 5 T - 6 T^{2} - 185 T^{3} - 739 T^{4} - 185 p T^{5} - 6 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
37 \( ( 1 - 7 T + 12 T^{2} + 175 T^{3} - 1669 T^{4} + 175 p T^{5} + 12 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
41 \( ( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} )^{2} \)
43 \( ( 1 - p T^{2} )^{8} \)
47 \( ( 1 - 12 T + 97 T^{2} - 600 T^{3} + 2641 T^{4} - 600 p T^{5} + 97 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )( 1 + 12 T + 97 T^{2} + 600 T^{3} + 2641 T^{4} + 600 p T^{5} + 97 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} ) \)
53 \( ( 1 - 6 T - 17 T^{2} + 420 T^{3} - 1619 T^{4} + 420 p T^{5} - 17 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )( 1 + 6 T - 17 T^{2} - 420 T^{3} - 1619 T^{4} - 420 p T^{5} - 17 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} ) \)
59 \( ( 1 - 15 T + 166 T^{2} - 1605 T^{3} + 14281 T^{4} - 1605 p T^{5} + 166 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} )( 1 + 15 T + 166 T^{2} + 1605 T^{3} + 14281 T^{4} + 1605 p T^{5} + 166 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} ) \)
61 \( ( 1 + p T^{2} + p^{2} T^{4} + p^{3} T^{6} + p^{4} T^{8} )^{2} \)
67 \( ( 1 + 13 T + p T^{2} )^{8} \)
71 \( ( 1 - 3 T - 62 T^{2} + 399 T^{3} + 3205 T^{4} + 399 p T^{5} - 62 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} )( 1 + 3 T - 62 T^{2} - 399 T^{3} + 3205 T^{4} - 399 p T^{5} - 62 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} ) \)
73 \( ( 1 + p T^{2} + p^{2} T^{4} + p^{3} T^{6} + p^{4} T^{8} )^{2} \)
79 \( ( 1 + p T^{2} + p^{2} T^{4} + p^{3} T^{6} + p^{4} T^{8} )^{2} \)
83 \( ( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} )^{2} \)
89 \( ( 1 - 9 T + p T^{2} )^{4}( 1 + 9 T + p T^{2} )^{4} \)
97 \( ( 1 + 17 T + 192 T^{2} + 1615 T^{3} + 8831 T^{4} + 1615 p T^{5} + 192 p^{2} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
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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

−5.07967363260263247125962434011, −4.88452191871450324022881804082, −4.84627490187079519612064120089, −4.64080829661820526060310982090, −4.42517317510318005073199156116, −4.33423142860740175041828266427, −4.32426539416367784376703319195, −4.12774075435553121452672700218, −3.86653699915197537433462811456, −3.81703828548082123110047151839, −3.60821845273483581345198959887, −3.14335217985389202799248693957, −3.03494126603669122292874903642, −3.03316022938755799293020086027, −2.89585754527829819190238181197, −2.68348368065515614122209828544, −2.63469280665277238941875095205, −2.44952051215274306660171749795, −1.82600190367722096371917388635, −1.70525901996948540872432637548, −1.61149509579184896117696705702, −1.58409347437661828091341627436, −1.54085199619496552913239883039, −0.824933596876542674844856619446, −0.29613251775007733020455175108, 0.29613251775007733020455175108, 0.824933596876542674844856619446, 1.54085199619496552913239883039, 1.58409347437661828091341627436, 1.61149509579184896117696705702, 1.70525901996948540872432637548, 1.82600190367722096371917388635, 2.44952051215274306660171749795, 2.63469280665277238941875095205, 2.68348368065515614122209828544, 2.89585754527829819190238181197, 3.03316022938755799293020086027, 3.03494126603669122292874903642, 3.14335217985389202799248693957, 3.60821845273483581345198959887, 3.81703828548082123110047151839, 3.86653699915197537433462811456, 4.12774075435553121452672700218, 4.32426539416367784376703319195, 4.33423142860740175041828266427, 4.42517317510318005073199156116, 4.64080829661820526060310982090, 4.84627490187079519612064120089, 4.88452191871450324022881804082, 5.07967363260263247125962434011

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

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