Properties

Label 16-48e16-1.1-c1e8-0-10
Degree $16$
Conductor $7.941\times 10^{26}$
Sign $1$
Analytic cond. $1.31243\times 10^{10}$
Root an. cond. $4.28923$
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  − 8·13-s − 40·37-s + 56·61-s + 8·109-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 32·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯
L(s)  = 1  − 2.21·13-s − 6.57·37-s + 7.17·61-s + 0.766·109-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2.46·169-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 + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + 0.0655·233-s + 0.0646·239-s + ⋯

Functional equation

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

Invariants

Degree: \(16\)
Conductor: \(2^{64} \cdot 3^{16}\)
Sign: $1$
Analytic conductor: \(1.31243\times 10^{10}\)
Root analytic conductor: \(4.28923\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{64} \cdot 3^{16} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.182008024\)
\(L(\frac12)\) \(\approx\) \(1.182008024\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( ( 1 + p^{2} T^{4} )^{4} \)
7 \( ( 1 - 94 T^{4} + p^{4} T^{8} )^{2} \)
11 \( ( 1 + p^{2} T^{4} )^{4} \)
13 \( ( 1 + 2 T + p T^{2} )^{4}( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \)
17 \( ( 1 + p T^{2} )^{8} \)
19 \( ( 1 - 46 T^{4} + p^{4} T^{8} )^{2} \)
23 \( ( 1 - p T^{2} )^{8} \)
29 \( ( 1 + p^{2} T^{4} )^{4} \)
31 \( ( 1 + 194 T^{4} + p^{4} T^{8} )^{2} \)
37 \( ( 1 + 10 T + p T^{2} )^{4}( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \)
41 \( ( 1 - p T^{2} )^{8} \)
43 \( ( 1 - 3214 T^{4} + p^{4} T^{8} )^{2} \)
47 \( ( 1 + p T^{2} )^{8} \)
53 \( ( 1 + p^{2} T^{4} )^{4} \)
59 \( ( 1 + p^{2} T^{4} )^{4} \)
61 \( ( 1 - 14 T + p T^{2} )^{4}( 1 + 74 T^{2} + p^{2} T^{4} )^{2} \)
67 \( ( 1 + 5906 T^{4} + p^{4} T^{8} )^{2} \)
71 \( ( 1 - p T^{2} )^{8} \)
73 \( ( 1 - 10 T + p T^{2} )^{4}( 1 + 10 T + p T^{2} )^{4} \)
79 \( ( 1 + 7682 T^{4} + p^{4} T^{8} )^{2} \)
83 \( ( 1 + p^{2} T^{4} )^{4} \)
89 \( ( 1 - p T^{2} )^{8} \)
97 \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{4} \)
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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

−3.69083816836246991729782461351, −3.66086313766565986499931004719, −3.55747878636090560139366885604, −3.38713189227330728940365645076, −3.36356506937563210708842703101, −3.11695992451155046724732611688, −3.11123595230006525351937570247, −2.85478079791618849232112689176, −2.74324972015038842783179014609, −2.72182985572843152092706780555, −2.36838316072915848759295974398, −2.32945836844049919345298260651, −2.24941842861742935569431297869, −2.06341049435003488305893124437, −2.03001565654541638454612306146, −2.00625974082682087726656183008, −1.74938485895175992015958681914, −1.49320818653316695854473762222, −1.34617278001121868603982789594, −1.16437911773182710116409593884, −1.14062921116334031518339444598, −0.65866564701460171188019754269, −0.65037999965081196962363681945, −0.20758499966174166311605329783, −0.20420525889157405788573879700, 0.20420525889157405788573879700, 0.20758499966174166311605329783, 0.65037999965081196962363681945, 0.65866564701460171188019754269, 1.14062921116334031518339444598, 1.16437911773182710116409593884, 1.34617278001121868603982789594, 1.49320818653316695854473762222, 1.74938485895175992015958681914, 2.00625974082682087726656183008, 2.03001565654541638454612306146, 2.06341049435003488305893124437, 2.24941842861742935569431297869, 2.32945836844049919345298260651, 2.36838316072915848759295974398, 2.72182985572843152092706780555, 2.74324972015038842783179014609, 2.85478079791618849232112689176, 3.11123595230006525351937570247, 3.11695992451155046724732611688, 3.36356506937563210708842703101, 3.38713189227330728940365645076, 3.55747878636090560139366885604, 3.66086313766565986499931004719, 3.69083816836246991729782461351

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

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