Properties

Label 16-1152e8-1.1-c1e8-0-4
Degree $16$
Conductor $3.102\times 10^{24}$
Sign $1$
Analytic cond. $5.12668\times 10^{7}$
Root an. cond. $3.03294$
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  + 8·19-s − 24·31-s − 16·37-s + 24·43-s + 24·49-s − 16·61-s − 32·67-s + 56·79-s + 32·109-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯
L(s)  = 1  + 1.83·19-s − 4.31·31-s − 2.63·37-s + 3.65·43-s + 24/7·49-s − 2.04·61-s − 3.90·67-s + 6.30·79-s + 3.06·109-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 + 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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{56} \cdot 3^{16}\)
Sign: $1$
Analytic conductor: \(5.12668\times 10^{7}\)
Root analytic conductor: \(3.03294\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{56} \cdot 3^{16} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(3.883859040\)
\(L(\frac12)\) \(\approx\) \(3.883859040\)
\(L(\frac{3}{2})\) 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 \( 1 - 12 T^{4} - 506 T^{8} - 12 p^{4} T^{12} + p^{8} T^{16} \)
7 \( ( 1 - 12 T^{2} + 106 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
11 \( 1 - 60 T^{4} - 14618 T^{8} - 60 p^{4} T^{12} + p^{8} T^{16} \)
13 \( ( 1 - 194 T^{4} + p^{4} T^{8} )^{2} \)
17 \( ( 1 + 28 T^{2} + 662 T^{4} + 28 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
19 \( ( 1 - 4 T + 8 T^{2} - 28 T^{3} - 46 T^{4} - 28 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
23 \( ( 1 - 12 T^{2} + 646 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
29 \( 1 + 180 T^{4} - 772538 T^{8} + 180 p^{4} T^{12} + p^{8} T^{16} \)
31 \( ( 1 + 6 T + 64 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{4} \)
37 \( ( 1 + 8 T + 32 T^{2} + 248 T^{3} + 1886 T^{4} + 248 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
41 \( ( 1 - 60 T^{2} + 4150 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
43 \( ( 1 - 12 T + 72 T^{2} - 564 T^{3} + 4402 T^{4} - 564 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
47 \( ( 1 - 20 T^{2} + 4070 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
53 \( 1 - 5772 T^{4} + 23807110 T^{8} - 5772 p^{4} T^{12} + p^{8} T^{16} \)
59 \( 1 - 7452 T^{4} + 27767206 T^{8} - 7452 p^{4} T^{12} + p^{8} T^{16} \)
61 \( ( 1 + 8 T + 32 T^{2} + 440 T^{3} + 6014 T^{4} + 440 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
67 \( ( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{4} \)
71 \( ( 1 - 92 T^{2} + 5030 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
73 \( ( 1 - 228 T^{2} + 23206 T^{4} - 228 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
79 \( ( 1 - 14 T + 200 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{4} \)
83 \( 1 + 20100 T^{4} + 185294374 T^{8} + 20100 p^{4} T^{12} + p^{8} T^{16} \)
89 \( ( 1 - 260 T^{2} + 30950 T^{4} - 260 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
97 \( ( 1 + 82 T^{2} + p^{2} T^{4} )^{4} \)
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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

−4.08845189282429991079967069240, −4.01006401473246068063172038379, −3.99183524117120684087988882506, −3.67211788129810419324161558776, −3.61494759586195468539558278528, −3.51012381257332033258149728583, −3.44096332639041984133794573081, −3.43872942457416310861972626210, −3.20692335155182357171492526169, −2.99595667346557616200554501331, −2.90177950710938501959687439580, −2.60756449431029740257920822735, −2.47946067490869324333941794392, −2.35504345765241218341822881802, −2.24744782978666888237340863287, −2.16259026528002634596770726423, −2.03562898235546417008442286797, −1.72454084157478137128531840367, −1.41457787333988374001295658614, −1.36299635298279026484515446661, −1.35578242007852919516161543336, −1.09372773694661418145322600136, −0.59687165222954820344282497790, −0.53683608020048707484273994794, −0.25985393226455843705158104790, 0.25985393226455843705158104790, 0.53683608020048707484273994794, 0.59687165222954820344282497790, 1.09372773694661418145322600136, 1.35578242007852919516161543336, 1.36299635298279026484515446661, 1.41457787333988374001295658614, 1.72454084157478137128531840367, 2.03562898235546417008442286797, 2.16259026528002634596770726423, 2.24744782978666888237340863287, 2.35504345765241218341822881802, 2.47946067490869324333941794392, 2.60756449431029740257920822735, 2.90177950710938501959687439580, 2.99595667346557616200554501331, 3.20692335155182357171492526169, 3.43872942457416310861972626210, 3.44096332639041984133794573081, 3.51012381257332033258149728583, 3.61494759586195468539558278528, 3.67211788129810419324161558776, 3.99183524117120684087988882506, 4.01006401473246068063172038379, 4.08845189282429991079967069240

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

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