Properties

Label 16-1216e8-1.1-c1e8-0-9
Degree $16$
Conductor $4.780\times 10^{24}$
Sign $1$
Analytic cond. $7.90106\times 10^{7}$
Root an. cond. $3.11605$
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  − 10·9-s + 40·17-s + 26·25-s − 12·49-s − 24·73-s + 43·81-s − 32·89-s + 24·97-s − 72·113-s + 54·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 400·153-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯
L(s)  = 1  − 3.33·9-s + 9.70·17-s + 26/5·25-s − 1.71·49-s − 2.80·73-s + 43/9·81-s − 3.39·89-s + 2.43·97-s − 6.77·113-s + 4.90·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 32.3·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2/13·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{48} \cdot 19^{8}\)
Sign: $1$
Analytic conductor: \(7.90106\times 10^{7}\)
Root analytic conductor: \(3.11605\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1216} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{48} \cdot 19^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(10.42244887\)
\(L(\frac12)\) \(\approx\) \(10.42244887\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( ( 1 + T^{2} )^{4} \)
good3 \( ( 1 + 5 T^{2} + 16 T^{4} + 5 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
5 \( ( 1 - 13 T^{2} + 84 T^{4} - 13 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
7 \( ( 1 + 3 T^{2} + p^{2} T^{4} )^{4} \)
11 \( ( 1 - 27 T^{2} + 416 T^{4} - 27 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
13 \( ( 1 - T^{2} + 264 T^{4} - p^{2} T^{6} + p^{4} T^{8} )^{2} \)
17 \( ( 1 - 5 T + p T^{2} )^{8} \)
23 \( ( 1 + 85 T^{2} + 2856 T^{4} + 85 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
29 \( ( 1 - 73 T^{2} + 2808 T^{4} - 73 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
31 \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{4} \)
37 \( ( 1 - 36 T^{2} + 950 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
41 \( ( 1 + p T^{2} )^{8} \)
43 \( ( 1 - 71 T^{2} + 3564 T^{4} - 71 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
47 \( ( 1 + 101 T^{2} + 5112 T^{4} + 101 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
53 \( ( 1 - 85 T^{2} + 6756 T^{4} - 85 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
59 \( ( 1 - 215 T^{2} + 18444 T^{4} - 215 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
61 \( ( 1 - 13 T^{2} - 1500 T^{4} - 13 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
67 \( ( 1 - 191 T^{2} + 17100 T^{4} - 191 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
71 \( ( 1 + 98 T^{2} + p^{2} T^{4} )^{4} \)
73 \( ( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{4} \)
79 \( ( 1 + 132 T^{2} + 8390 T^{4} + 132 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
83 \( ( 1 - 102 T^{2} + p^{2} T^{4} )^{4} \)
89 \( ( 1 + 8 T + 62 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{4} \)
97 \( ( 1 - 6 T + 170 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{4} \)
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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

−4.02578017180304037392457850052, −3.86819229434019105338563217548, −3.80768512251790735407297090530, −3.55694211544024947910049892385, −3.49248163063783348310902555142, −3.38000801168883427943071228047, −3.34262500070991929730988144828, −3.13392879742463735289517653005, −3.09818265632265382456473813380, −2.94361977816303228611969455357, −2.93504849566132738532662159349, −2.87994764277527998292431646928, −2.74784087615990220503425551302, −2.57667518255882584701265560532, −2.37916494402469497161880132543, −2.07375692236326742150565582249, −1.78878215865209644962971452947, −1.52143263540541849284876611229, −1.39862032517824890457937583829, −1.29382277824264498954001332178, −1.21974153605084793001497723306, −0.962322197851378193396681577292, −0.843841788757877423698514414155, −0.59846594733550546656956410950, −0.33758132492753181000324643856, 0.33758132492753181000324643856, 0.59846594733550546656956410950, 0.843841788757877423698514414155, 0.962322197851378193396681577292, 1.21974153605084793001497723306, 1.29382277824264498954001332178, 1.39862032517824890457937583829, 1.52143263540541849284876611229, 1.78878215865209644962971452947, 2.07375692236326742150565582249, 2.37916494402469497161880132543, 2.57667518255882584701265560532, 2.74784087615990220503425551302, 2.87994764277527998292431646928, 2.93504849566132738532662159349, 2.94361977816303228611969455357, 3.09818265632265382456473813380, 3.13392879742463735289517653005, 3.34262500070991929730988144828, 3.38000801168883427943071228047, 3.49248163063783348310902555142, 3.55694211544024947910049892385, 3.80768512251790735407297090530, 3.86819229434019105338563217548, 4.02578017180304037392457850052

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

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