Properties

Label 16-2e72-1.1-c1e8-0-1
Degree $16$
Conductor $4.722\times 10^{21}$
Sign $1$
Analytic cond. $78050.6$
Root an. cond. $2.02196$
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  + 8·5-s − 8·13-s + 32·25-s + 8·29-s − 40·37-s + 24·49-s − 8·53-s − 40·61-s − 64·65-s + 12·81-s + 64·97-s − 40·101-s + 40·109-s + 16·113-s + 88·125-s + 127-s + 131-s + 137-s + 139-s + 64·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 32·169-s + 173-s + ⋯
L(s)  = 1  + 3.57·5-s − 2.21·13-s + 32/5·25-s + 1.48·29-s − 6.57·37-s + 24/7·49-s − 1.09·53-s − 5.12·61-s − 7.93·65-s + 4/3·81-s + 6.49·97-s − 3.98·101-s + 3.83·109-s + 1.50·113-s + 7.87·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 5.31·145-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{72}\)
Sign: $1$
Analytic conductor: \(78050.6\)
Root analytic conductor: \(2.02196\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{512} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{72} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(3.265107670\)
\(L(\frac12)\) \(\approx\) \(3.265107670\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
good3 \( 1 - 4 p T^{4} + 70 T^{8} - 4 p^{5} T^{12} + p^{8} T^{16} \)
5 \( ( 1 - 4 T + 8 T^{2} - 12 T^{3} + 14 T^{4} - 12 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
7 \( ( 1 - 12 T^{2} + 102 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
11 \( 1 - 460 T^{4} + 82054 T^{8} - 460 p^{4} T^{12} + p^{8} T^{16} \)
13 \( ( 1 + 4 T + 8 T^{2} + 44 T^{3} + 238 T^{4} + 44 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
17 \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{4} \)
19 \( 1 - 12 T^{4} - 247354 T^{8} - 12 p^{4} T^{12} + p^{8} T^{16} \)
23 \( ( 1 - 12 T^{2} + 1062 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
29 \( ( 1 - 4 T + 8 T^{2} + 20 T^{3} - 1106 T^{4} + 20 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
31 \( ( 1 + 60 T^{2} + 2310 T^{4} + 60 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
37 \( ( 1 + 20 T + 200 T^{2} + 1660 T^{3} + 11662 T^{4} + 1660 p T^{5} + 200 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
41 \( ( 1 - 66 T^{2} + p^{2} T^{4} )^{4} \)
43 \( 1 + 6068 T^{4} + 15827590 T^{8} + 6068 p^{4} T^{12} + p^{8} T^{16} \)
47 \( ( 1 + 124 T^{2} + 7750 T^{4} + 124 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
53 \( ( 1 + 4 T + 8 T^{2} + 76 T^{3} - 434 T^{4} + 76 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
59 \( 1 + 12084 T^{4} + 60324614 T^{8} + 12084 p^{4} T^{12} + p^{8} T^{16} \)
61 \( ( 1 + 20 T + 200 T^{2} + 1500 T^{3} + 11054 T^{4} + 1500 p T^{5} + 200 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
67 \( 1 + 116 T^{4} - 21877946 T^{8} + 116 p^{4} T^{12} + p^{8} T^{16} \)
71 \( ( 1 - 76 T^{2} + 9958 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
73 \( ( 1 - 140 T^{2} + 14406 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
79 \( ( 1 + 60 T^{2} + 5190 T^{4} + 60 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
83 \( 1 - 3468 T^{4} - 36032314 T^{8} - 3468 p^{4} T^{12} + p^{8} T^{16} \)
89 \( ( 1 - 332 T^{2} + 43270 T^{4} - 332 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
97 \( ( 1 - 16 T + 250 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{4} \)
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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

−4.88620447094572529493857703131, −4.75178476715400533185907338332, −4.57342866015957090660895347254, −4.52100256636222113202184407758, −4.42523014947871481965286996158, −4.13351153626627673673790460807, −3.84152083693077467313520762439, −3.78897788530362012449288083232, −3.53704733589736413643395014094, −3.35455143084992229620120561989, −3.17065645868108135610550623055, −3.16278139991725190232395647040, −2.95660436528853061084817827584, −2.76122460026501772446708132863, −2.73269470059562279233877530719, −2.37498512763698241093886496167, −2.04833352019253287965533360857, −2.04500901582175209608292805278, −1.95666886106626110071971252243, −1.87131517503645077205991377619, −1.69610534499010123465804651596, −1.50307546148275386429436592195, −0.952473800402299021182891169430, −0.887355469493061669381303777417, −0.24354278763527971365171419129, 0.24354278763527971365171419129, 0.887355469493061669381303777417, 0.952473800402299021182891169430, 1.50307546148275386429436592195, 1.69610534499010123465804651596, 1.87131517503645077205991377619, 1.95666886106626110071971252243, 2.04500901582175209608292805278, 2.04833352019253287965533360857, 2.37498512763698241093886496167, 2.73269470059562279233877530719, 2.76122460026501772446708132863, 2.95660436528853061084817827584, 3.16278139991725190232395647040, 3.17065645868108135610550623055, 3.35455143084992229620120561989, 3.53704733589736413643395014094, 3.78897788530362012449288083232, 3.84152083693077467313520762439, 4.13351153626627673673790460807, 4.42523014947871481965286996158, 4.52100256636222113202184407758, 4.57342866015957090660895347254, 4.75178476715400533185907338332, 4.88620447094572529493857703131

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

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