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Properties

Label 16-672e8-1.1-c1e8-0-5
Degree $16$
Conductor $4.159\times 10^{22}$
Sign $1$
Analytic cond. $687339.$
Root an. cond. $2.31645$
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  − 8·3-s − 4·7-s + 36·9-s − 8·19-s + 32·21-s + 12·25-s − 120·27-s + 16·31-s + 8·37-s − 16·47-s + 8·49-s + 16·53-s + 64·57-s − 32·59-s − 144·63-s − 96·75-s + 330·81-s − 16·83-s − 128·93-s + 48·103-s − 64·111-s + 20·121-s + 127-s + 131-s + 32·133-s + 137-s + 139-s + ⋯
L(s)  = 1  − 4.61·3-s − 1.51·7-s + 12·9-s − 1.83·19-s + 6.98·21-s + 12/5·25-s − 23.0·27-s + 2.87·31-s + 1.31·37-s − 2.33·47-s + 8/7·49-s + 2.19·53-s + 8.47·57-s − 4.16·59-s − 18.1·63-s − 11.0·75-s + 36.6·81-s − 1.75·83-s − 13.2·93-s + 4.72·103-s − 6.07·111-s + 1.81·121-s + 0.0887·127-s + 0.0873·131-s + 2.77·133-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{40} \cdot 3^{8} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(687339.\)
Root analytic conductor: \(2.31645\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{40} \cdot 3^{8} \cdot 7^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.3142111602\)
\(L(\frac12)\) \(\approx\) \(0.3142111602\)
\(L(\frac{3}{2})\) 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 \( ( 1 + T )^{8} \)
7 \( 1 + 4 T + 8 T^{2} + 20 T^{3} + 46 T^{4} + 20 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
good5 \( 1 - 12 T^{2} + 56 T^{4} + 28 T^{6} - 1074 T^{8} + 28 p^{2} T^{10} + 56 p^{4} T^{12} - 12 p^{6} T^{14} + p^{8} T^{16} \)
11 \( 1 - 20 T^{2} + 296 T^{4} - 2268 T^{6} + 20718 T^{8} - 2268 p^{2} T^{10} + 296 p^{4} T^{12} - 20 p^{6} T^{14} + p^{8} T^{16} \)
13 \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{4} \)
17 \( 1 - 108 T^{2} + 5432 T^{4} - 166628 T^{6} + 3420078 T^{8} - 166628 p^{2} T^{10} + 5432 p^{4} T^{12} - 108 p^{6} T^{14} + p^{8} T^{16} \)
19 \( ( 1 + 4 T + 32 T^{2} + 132 T^{3} + 558 T^{4} + 132 p T^{5} + 32 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
23 \( 1 - 100 T^{2} + 5000 T^{4} - 174764 T^{6} + 4622926 T^{8} - 174764 p^{2} T^{10} + 5000 p^{4} T^{12} - 100 p^{6} T^{14} + p^{8} T^{16} \)
29 \( ( 1 + 36 T^{2} + 96 T^{3} + 390 T^{4} + 96 p T^{5} + 36 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
31 \( ( 1 - 8 T + 100 T^{2} - 616 T^{3} + 4534 T^{4} - 616 p T^{5} + 100 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
37 \( ( 1 - 4 T + 40 T^{2} + 292 T^{3} - 866 T^{4} + 292 p T^{5} + 40 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
41 \( 1 - 204 T^{2} + 21176 T^{4} - 1439172 T^{6} + 69360622 T^{8} - 1439172 p^{2} T^{10} + 21176 p^{4} T^{12} - 204 p^{6} T^{14} + p^{8} T^{16} \)
43 \( 1 - 192 T^{2} + 18396 T^{4} - 1162560 T^{6} + 55940774 T^{8} - 1162560 p^{2} T^{10} + 18396 p^{4} T^{12} - 192 p^{6} T^{14} + p^{8} T^{16} \)
47 \( ( 1 + 8 T + 108 T^{2} + 872 T^{3} + 6758 T^{4} + 872 p T^{5} + 108 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
53 \( ( 1 - 8 T + 132 T^{2} - 632 T^{3} + 7718 T^{4} - 632 p T^{5} + 132 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
59 \( ( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{4} \)
61 \( 1 - 184 T^{2} + 16284 T^{4} - 814280 T^{6} + 38728934 T^{8} - 814280 p^{2} T^{10} + 16284 p^{4} T^{12} - 184 p^{6} T^{14} + p^{8} T^{16} \)
67 \( 1 - 240 T^{2} + 37436 T^{4} - 3815056 T^{6} + 299829030 T^{8} - 3815056 p^{2} T^{10} + 37436 p^{4} T^{12} - 240 p^{6} T^{14} + p^{8} T^{16} \)
71 \( 1 - 484 T^{2} + 107912 T^{4} - 14382508 T^{6} + 1250123214 T^{8} - 14382508 p^{2} T^{10} + 107912 p^{4} T^{12} - 484 p^{6} T^{14} + p^{8} T^{16} \)
73 \( 1 - 408 T^{2} + 80380 T^{4} - 10037032 T^{6} + 869620166 T^{8} - 10037032 p^{2} T^{10} + 80380 p^{4} T^{12} - 408 p^{6} T^{14} + p^{8} T^{16} \)
79 \( 1 - 368 T^{2} + 73436 T^{4} - 9593744 T^{6} + 892827078 T^{8} - 9593744 p^{2} T^{10} + 73436 p^{4} T^{12} - 368 p^{6} T^{14} + p^{8} T^{16} \)
83 \( ( 1 + 8 T + 124 T^{2} + 1480 T^{3} + 7318 T^{4} + 1480 p T^{5} + 124 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
89 \( 1 + 52 T^{2} - 328 T^{4} + 280764 T^{6} + 120770670 T^{8} + 280764 p^{2} T^{10} - 328 p^{4} T^{12} + 52 p^{6} T^{14} + p^{8} T^{16} \)
97 \( 1 - 344 T^{2} + 58172 T^{4} - 6404840 T^{6} + 615495942 T^{8} - 6404840 p^{2} T^{10} + 58172 p^{4} T^{12} - 344 p^{6} T^{14} + p^{8} T^{16} \)
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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

−4.69662335493461982192036812010, −4.49144092549669342231905410746, −4.38822741116252278680589614782, −4.27143451512024032757198290480, −4.22902707303103262258035318695, −4.19377472299050687178511805027, −3.85216690492342651215088822389, −3.58737312768157205997531490504, −3.45749368577914291646528867317, −3.40310535952134666379349972430, −3.17182057126612112133580646691, −2.98482343889860408751376488430, −2.86159145697689466383963184149, −2.67802252048264656560045916424, −2.64542618365220786862307016810, −2.16220883412885156634864926429, −1.97777243901357943935704782484, −1.91013467085569052418010114310, −1.76458186678987297445594492216, −1.22741220500977443155452757343, −1.16556609990206262074365838610, −1.06123788719236089520345747959, −0.70149140976081297009624337784, −0.44272478481752538775293498285, −0.27147903248093502454737934707, 0.27147903248093502454737934707, 0.44272478481752538775293498285, 0.70149140976081297009624337784, 1.06123788719236089520345747959, 1.16556609990206262074365838610, 1.22741220500977443155452757343, 1.76458186678987297445594492216, 1.91013467085569052418010114310, 1.97777243901357943935704782484, 2.16220883412885156634864926429, 2.64542618365220786862307016810, 2.67802252048264656560045916424, 2.86159145697689466383963184149, 2.98482343889860408751376488430, 3.17182057126612112133580646691, 3.40310535952134666379349972430, 3.45749368577914291646528867317, 3.58737312768157205997531490504, 3.85216690492342651215088822389, 4.19377472299050687178511805027, 4.22902707303103262258035318695, 4.27143451512024032757198290480, 4.38822741116252278680589614782, 4.49144092549669342231905410746, 4.69662335493461982192036812010

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

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