Properties

Label 16-12e24-1.1-c2e8-0-13
Degree $16$
Conductor $7.950\times 10^{25}$
Sign $1$
Analytic cond. $2.41562\times 10^{13}$
Root an. cond. $6.86182$
Motivic weight $2$
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  + 16·13-s + 84·25-s − 32·37-s − 148·49-s − 96·61-s − 216·73-s − 40·97-s − 352·109-s + 900·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 920·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  + 1.23·13-s + 3.35·25-s − 0.864·37-s − 3.02·49-s − 1.57·61-s − 2.95·73-s − 0.412·97-s − 3.22·109-s + 7.43·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 5.44·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{48} \cdot 3^{24}\)
Sign: $1$
Analytic conductor: \(2.41562\times 10^{13}\)
Root analytic conductor: \(6.86182\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1728} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{48} \cdot 3^{24} ,\ ( \ : [1]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(5.376977357\)
\(L(\frac12)\) \(\approx\) \(5.376977357\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( ( 1 - 42 T^{2} + 1043 T^{4} - 42 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
7 \( ( 1 + 74 T^{2} + 2643 T^{4} + 74 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
11 \( ( 1 - 450 T^{2} + 79619 T^{4} - 450 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
13 \( ( 1 - 4 T + 270 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} )^{4} \)
17 \( ( 1 - 76 T^{2} + 127014 T^{4} - 76 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
19 \( ( 1 + 980 T^{2} + 459270 T^{4} + 980 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
23 \( ( 1 - 1684 T^{2} + 1227174 T^{4} - 1684 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
29 \( ( 1 - 2028 T^{2} + 2147846 T^{4} - 2028 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
31 \( ( 1 - 646 T^{2} - 355581 T^{4} - 646 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
37 \( ( 1 + 8 T + 954 T^{2} + 8 p^{2} T^{3} + p^{4} T^{4} )^{4} \)
41 \( ( 1 - 5076 T^{2} + 12019238 T^{4} - 5076 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
43 \( ( 1 + 1916 T^{2} + 7662054 T^{4} + 1916 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
47 \( ( 1 - 5052 T^{2} + 14107910 T^{4} - 5052 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
53 \( ( 1 - 2538 T^{2} + 14597075 T^{4} - 2538 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
59 \( ( 1 - 6492 T^{2} + 25440038 T^{4} - 6492 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
61 \( ( 1 + 24 T + 2978 T^{2} + 24 p^{2} T^{3} + p^{4} T^{4} )^{4} \)
67 \( ( 1 + 7676 T^{2} + 45068838 T^{4} + 7676 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
71 \( ( 1 - 5292 T^{2} + 24922406 T^{4} - 5292 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
73 \( ( 1 + 54 T + 2675 T^{2} + 54 p^{2} T^{3} + p^{4} T^{4} )^{4} \)
79 \( ( 1 + 11660 T^{2} + 68532390 T^{4} + 11660 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
83 \( ( 1 - 3634 T^{2} - 15003021 T^{4} - 3634 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
89 \( ( 1 + 7332 T^{2} + 109063046 T^{4} + 7332 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
97 \( ( 1 + 10 T + 14235 T^{2} + 10 p^{2} T^{3} + p^{4} T^{4} )^{4} \)
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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

−3.60908740425306774205478729141, −3.56577097622615782659030517623, −3.45408838278262594403478360418, −3.21106423073226073935993043075, −3.19184949983705986024491780085, −3.14187255023324989744091649832, −3.01267039141437514057938849058, −2.91145048071967505700216232786, −2.72108344323252623364356473497, −2.61495549885667781636987101025, −2.60117490676669736400016079068, −2.16319818224565710502461344198, −2.05513623801203631181633043330, −1.96188380626474678422537689022, −1.83759226778075460502412433281, −1.75598643556368236378983011989, −1.45433132206426546669395842613, −1.34848807309558043553702694788, −1.20199136471032813362627915931, −1.17231258139626588283554867587, −0.905458968056834117406876203321, −0.70077206306656760896448132070, −0.54129355513714927266377432032, −0.22444004685550114360863788622, −0.20046215263536593428537172277, 0.20046215263536593428537172277, 0.22444004685550114360863788622, 0.54129355513714927266377432032, 0.70077206306656760896448132070, 0.905458968056834117406876203321, 1.17231258139626588283554867587, 1.20199136471032813362627915931, 1.34848807309558043553702694788, 1.45433132206426546669395842613, 1.75598643556368236378983011989, 1.83759226778075460502412433281, 1.96188380626474678422537689022, 2.05513623801203631181633043330, 2.16319818224565710502461344198, 2.60117490676669736400016079068, 2.61495549885667781636987101025, 2.72108344323252623364356473497, 2.91145048071967505700216232786, 3.01267039141437514057938849058, 3.14187255023324989744091649832, 3.19184949983705986024491780085, 3.21106423073226073935993043075, 3.45408838278262594403478360418, 3.56577097622615782659030517623, 3.60908740425306774205478729141

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

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