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Properties

Label 16-315e8-1.1-c1e8-0-3
Degree $16$
Conductor $9.694\times 10^{19}$
Sign $1$
Analytic cond. $1602.14$
Root an. cond. $1.58596$
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  + 8·4-s + 2·9-s + 18·11-s + 24·16-s + 10·25-s + 16·36-s + 144·44-s − 14·49-s − 2·79-s − 15·81-s + 36·99-s + 80·100-s − 44·109-s + 149·121-s + 127-s + 131-s + 137-s + 139-s + 48·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 19·169-s + 173-s + 432·176-s + ⋯
L(s)  = 1  + 4·4-s + 2/3·9-s + 5.42·11-s + 6·16-s + 2·25-s + 8/3·36-s + 21.7·44-s − 2·49-s − 0.225·79-s − 5/3·81-s + 3.61·99-s + 8·100-s − 4.21·109-s + 13.5·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 4·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.46·169-s + 0.0760·173-s + 32.5·176-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(3^{16} \cdot 5^{8} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(1602.14\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 3^{16} \cdot 5^{8} \cdot 7^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(18.60994976\)
\(L(\frac12)\) \(\approx\) \(18.60994976\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( ( 1 - T^{2} + p^{2} T^{4} )^{2} \)
5 \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
7 \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \)
good2 \( ( 1 - p T^{2} + p^{2} T^{4} )^{4} \)
11 \( ( 1 - 3 T + p T^{2} )^{4}( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \)
13 \( ( 1 + 19 T^{2} + p^{2} T^{4} )^{2}( 1 - 19 T^{2} + 192 T^{4} - 19 p^{2} T^{6} + p^{4} T^{8} ) \)
17 \( ( 1 + 29 T^{2} + 552 T^{4} + 29 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
19 \( ( 1 - p T^{2} )^{8} \)
23 \( ( 1 - p T^{2} + p^{2} T^{4} )^{4} \)
29 \( ( 1 - 9 T + 52 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2}( 1 + 9 T + 52 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \)
31 \( ( 1 + p T^{2} + p^{2} T^{4} )^{4} \)
37 \( ( 1 - p T^{2} )^{8} \)
41 \( ( 1 - p T^{2} + p^{2} T^{4} )^{4} \)
43 \( ( 1 + p T^{2} + p^{2} T^{4} )^{4} \)
47 \( ( 1 + 31 T^{2} + p^{2} T^{4} )^{2}( 1 - 31 T^{2} - 1248 T^{4} - 31 p^{2} T^{6} + p^{4} T^{8} ) \)
53 \( ( 1 + p T^{2} )^{8} \)
59 \( ( 1 - p T^{2} + p^{2} T^{4} )^{4} \)
61 \( ( 1 + p T^{2} + p^{2} T^{4} )^{4} \)
67 \( ( 1 + p T^{2} + p^{2} T^{4} )^{4} \)
71 \( ( 1 - 12 T + 73 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2}( 1 + 12 T + 73 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
73 \( ( 1 - 34 T^{2} - 4173 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
79 \( ( 1 + T + p T^{2} )^{4}( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \)
83 \( ( 1 - 86 T^{2} + p^{2} T^{4} )^{2}( 1 + 86 T^{2} + 507 T^{4} + 86 p^{2} T^{6} + p^{4} T^{8} ) \)
89 \( ( 1 + p T^{2} )^{8} \)
97 \( ( 1 - 149 T^{2} + p^{2} T^{4} )^{2}( 1 + 149 T^{2} + 12792 T^{4} + 149 p^{2} T^{6} + p^{4} T^{8} ) \)
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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

−5.38710147250028367880582974058, −4.95018179885979440676735078639, −4.78351120858332111524808298344, −4.75776657264465266181801382952, −4.54638053923673442310430910499, −4.50504657747136154729904136292, −4.21352486763969052407870249470, −4.18140609304991020353601269906, −4.03557493646433353737937445329, −3.73659200235889966652948328189, −3.63013974020717871755136674489, −3.47248947286611852944568715519, −3.27010955502497572455473284230, −3.08052970798113928929772017298, −3.01126446833456109881175090711, −2.58989377110210238494165120036, −2.54433216816027683592805582607, −2.51679075134356188061877534772, −2.14999361342721846717754173268, −1.74263349474675263328206527857, −1.65324906473425603656228003963, −1.58823519290951605489025979166, −1.44959440737507015734992759619, −1.25811067838583744828351335100, −0.837531570709513230772019113474, 0.837531570709513230772019113474, 1.25811067838583744828351335100, 1.44959440737507015734992759619, 1.58823519290951605489025979166, 1.65324906473425603656228003963, 1.74263349474675263328206527857, 2.14999361342721846717754173268, 2.51679075134356188061877534772, 2.54433216816027683592805582607, 2.58989377110210238494165120036, 3.01126446833456109881175090711, 3.08052970798113928929772017298, 3.27010955502497572455473284230, 3.47248947286611852944568715519, 3.63013974020717871755136674489, 3.73659200235889966652948328189, 4.03557493646433353737937445329, 4.18140609304991020353601269906, 4.21352486763969052407870249470, 4.50504657747136154729904136292, 4.54638053923673442310430910499, 4.75776657264465266181801382952, 4.78351120858332111524808298344, 4.95018179885979440676735078639, 5.38710147250028367880582974058

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

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