Properties

Label 16-48e16-1.1-c2e8-0-8
Degree $16$
Conductor $7.941\times 10^{26}$
Sign $1$
Analytic cond. $2.41290\times 10^{14}$
Root an. cond. $7.92334$
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  + 80·25-s + 128·31-s − 184·49-s + 160·73-s + 384·79-s − 192·97-s + 896·103-s + 360·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 408·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯
L(s)  = 1  + 16/5·25-s + 4.12·31-s − 3.75·49-s + 2.19·73-s + 4.86·79-s − 1.97·97-s + 8.69·103-s + 2.97·121-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 − 2.41·169-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 + 0.00448·223-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{64} \cdot 3^{16}\)
Sign: $1$
Analytic conductor: \(2.41290\times 10^{14}\)
Root analytic conductor: \(7.92334\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2304} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{64} \cdot 3^{16} ,\ ( \ : [1]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(33.64923390\)
\(L(\frac12)\) \(\approx\) \(33.64923390\)
\(L(2)\) 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 \( ( 1 - 8 p T^{2} + 818 T^{4} - 8 p^{5} T^{6} + p^{8} T^{8} )^{2} \)
7 \( ( 1 + 46 T^{2} + p^{4} T^{4} )^{4} \)
11 \( ( 1 - 180 T^{2} + 24070 T^{4} - 180 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
13 \( ( 1 + 204 T^{2} + 14278 T^{4} + 204 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
17 \( ( 1 - 320 T^{2} + 189314 T^{4} - 320 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
19 \( ( 1 - 60 T^{2} + 48550 T^{4} - 60 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
23 \( ( 1 - 1652 T^{2} + 1188710 T^{4} - 1652 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
29 \( ( 1 - 2376 T^{2} + 2685298 T^{4} - 2376 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
31 \( ( 1 - 32 T + 1710 T^{2} - 32 p^{2} T^{3} + p^{4} T^{4} )^{4} \)
37 \( ( 1 + 3972 T^{2} + 7479526 T^{4} + 3972 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
41 \( ( 1 - 5792 T^{2} + 13955138 T^{4} - 5792 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
43 \( ( 1 + 996 T^{2} + 6872614 T^{4} + 996 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
47 \( ( 1 - 4660 T^{2} + 10875174 T^{4} - 4660 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
53 \( ( 1 - 8712 T^{2} + 34754866 T^{4} - 8712 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
59 \( ( 1 - 9316 T^{2} + 45079718 T^{4} - 9316 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
61 \( ( 1 + 9412 T^{2} + 45525030 T^{4} + 9412 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
67 \( ( 1 + 12484 T^{2} + 74951718 T^{4} + 12484 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
71 \( ( 1 - 9076 T^{2} + 54164454 T^{4} - 9076 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
73 \( ( 1 - 40 T + 7730 T^{2} - 40 p^{2} T^{3} + p^{4} T^{4} )^{4} \)
79 \( ( 1 - 96 T + 8494 T^{2} - 96 p^{2} T^{3} + p^{4} T^{4} )^{4} \)
83 \( ( 1 - 24436 T^{2} + 241946438 T^{4} - 24436 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
89 \( ( 1 - 25856 T^{2} + 284297666 T^{4} - 25856 p^{4} T^{6} + p^{8} T^{8} )^{2} \)
97 \( ( 1 + 48 T + 18562 T^{2} + 48 p^{2} T^{3} + p^{4} T^{4} )^{4} \)
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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

−3.58447835358054931255767912615, −3.22672599309104584992815242368, −3.17730026051721510941217992380, −3.16809653777916044784268004572, −3.08677628830640294213885569231, −3.07210495179724402693426781396, −2.91990728933810983430627366535, −2.90305705142305113887467483286, −2.49221465280168300728744226964, −2.48931387189949326639992150344, −2.16624527872641957815939590950, −2.08292591629591784892860141045, −2.04575244351228684606999789139, −2.01979870992858033810914043333, −1.81899064377171333151863695770, −1.60734898557946565666905116229, −1.55579388305402094976728899450, −1.25041416262631540497005604027, −0.957417458927080583062625732453, −0.820434853989070251230620953222, −0.77625161654732989501787650969, −0.71786232759551217894640169585, −0.64028772511138999182576846039, −0.50401384548322771791248873724, −0.21045616311403144164334553071, 0.21045616311403144164334553071, 0.50401384548322771791248873724, 0.64028772511138999182576846039, 0.71786232759551217894640169585, 0.77625161654732989501787650969, 0.820434853989070251230620953222, 0.957417458927080583062625732453, 1.25041416262631540497005604027, 1.55579388305402094976728899450, 1.60734898557946565666905116229, 1.81899064377171333151863695770, 2.01979870992858033810914043333, 2.04575244351228684606999789139, 2.08292591629591784892860141045, 2.16624527872641957815939590950, 2.48931387189949326639992150344, 2.49221465280168300728744226964, 2.90305705142305113887467483286, 2.91990728933810983430627366535, 3.07210495179724402693426781396, 3.08677628830640294213885569231, 3.16809653777916044784268004572, 3.17730026051721510941217992380, 3.22672599309104584992815242368, 3.58447835358054931255767912615

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

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