Properties

Label 16-1840e8-1.1-c1e8-0-1
Degree $16$
Conductor $1.314\times 10^{26}$
Sign $1$
Analytic cond. $2.17149\times 10^{9}$
Root an. cond. $3.83307$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 6·5-s + 8·9-s − 4·11-s − 8·19-s + 10·25-s − 8·29-s − 16·41-s − 48·45-s + 28·49-s + 24·55-s − 16·61-s + 48·71-s + 48·79-s + 30·81-s + 16·89-s + 48·95-s − 32·99-s − 24·101-s − 8·109-s − 44·121-s + 14·125-s + 127-s + 131-s + 137-s + 139-s + 48·145-s + 149-s + ⋯
L(s)  = 1  − 2.68·5-s + 8/3·9-s − 1.20·11-s − 1.83·19-s + 2·25-s − 1.48·29-s − 2.49·41-s − 7.15·45-s + 4·49-s + 3.23·55-s − 2.04·61-s + 5.69·71-s + 5.40·79-s + 10/3·81-s + 1.69·89-s + 4.92·95-s − 3.21·99-s − 2.38·101-s − 0.766·109-s − 4·121-s + 1.25·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 3.98·145-s + 0.0819·149-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{8} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{8} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{32} \cdot 5^{8} \cdot 23^{8}\)
Sign: $1$
Analytic conductor: \(2.17149\times 10^{9}\)
Root analytic conductor: \(3.83307\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1840} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{32} \cdot 5^{8} \cdot 23^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.2785569766\)
\(L(\frac12)\) \(\approx\) \(0.2785569766\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + 6 T + 26 T^{2} + 82 T^{3} + 206 T^{4} + 82 p T^{5} + 26 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
23 \( ( 1 + T^{2} )^{4} \)
good3 \( 1 - 8 T^{2} + 34 T^{4} - 32 p T^{6} + 259 T^{8} - 32 p^{3} T^{10} + 34 p^{4} T^{12} - 8 p^{6} T^{14} + p^{8} T^{16} \)
7 \( 1 - 4 p T^{2} + 348 T^{4} - 2708 T^{6} + 18230 T^{8} - 2708 p^{2} T^{10} + 348 p^{4} T^{12} - 4 p^{7} T^{14} + p^{8} T^{16} \)
11 \( ( 1 + 2 T + 28 T^{2} + 2 p T^{3} + 346 T^{4} + 2 p^{2} T^{5} + 28 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
13 \( 1 - 40 T^{2} + 914 T^{4} - 15968 T^{6} + 228595 T^{8} - 15968 p^{2} T^{10} + 914 p^{4} T^{12} - 40 p^{6} T^{14} + p^{8} T^{16} \)
17 \( 1 - 56 T^{2} + 1544 T^{4} - 36168 T^{6} + 720894 T^{8} - 36168 p^{2} T^{10} + 1544 p^{4} T^{12} - 56 p^{6} T^{14} + p^{8} T^{16} \)
19 \( ( 1 + 4 T + 70 T^{2} + 200 T^{3} + 1918 T^{4} + 200 p T^{5} + 70 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
29 \( ( 1 + 4 T + 94 T^{2} + 344 T^{3} + 3775 T^{4} + 344 p T^{5} + 94 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
31 \( ( 1 + 50 T^{2} + 256 T^{3} + 1011 T^{4} + 256 p T^{5} + 50 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
37 \( 1 - 4 p T^{2} + 11228 T^{4} - 572828 T^{6} + 23118550 T^{8} - 572828 p^{2} T^{10} + 11228 p^{4} T^{12} - 4 p^{7} T^{14} + p^{8} T^{16} \)
41 \( ( 1 + 8 T + 70 T^{2} + 616 T^{3} + 4863 T^{4} + 616 p T^{5} + 70 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
43 \( 1 - 92 T^{2} + 8096 T^{4} - 11164 p T^{6} + 22994446 T^{8} - 11164 p^{3} T^{10} + 8096 p^{4} T^{12} - 92 p^{6} T^{14} + p^{8} T^{16} \)
47 \( 1 - 96 T^{2} + 11626 T^{4} - 665136 T^{6} + 41700939 T^{8} - 665136 p^{2} T^{10} + 11626 p^{4} T^{12} - 96 p^{6} T^{14} + p^{8} T^{16} \)
53 \( 1 - 200 T^{2} + 23516 T^{4} - 1842744 T^{6} + 111792678 T^{8} - 1842744 p^{2} T^{10} + 23516 p^{4} T^{12} - 200 p^{6} T^{14} + p^{8} T^{16} \)
59 \( ( 1 + 216 T^{2} + 16 T^{3} + 18542 T^{4} + 16 p T^{5} + 216 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
61 \( ( 1 + 8 T + 142 T^{2} + 316 T^{3} + 7126 T^{4} + 316 p T^{5} + 142 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
67 \( 1 - 516 T^{2} + 117772 T^{4} - 15527980 T^{6} + 1293477494 T^{8} - 15527980 p^{2} T^{10} + 117772 p^{4} T^{12} - 516 p^{6} T^{14} + p^{8} T^{16} \)
71 \( ( 1 - 24 T + 350 T^{2} - 3544 T^{3} + 32183 T^{4} - 3544 p T^{5} + 350 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
73 \( 1 - 216 T^{2} + 38578 T^{4} - 4059072 T^{6} + 362257971 T^{8} - 4059072 p^{2} T^{10} + 38578 p^{4} T^{12} - 216 p^{6} T^{14} + p^{8} T^{16} \)
79 \( ( 1 - 24 T + 494 T^{2} - 6100 T^{3} + 65598 T^{4} - 6100 p T^{5} + 494 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
83 \( 1 - 88 T^{2} + 20552 T^{4} - 1491272 T^{6} + 193546270 T^{8} - 1491272 p^{2} T^{10} + 20552 p^{4} T^{12} - 88 p^{6} T^{14} + p^{8} T^{16} \)
89 \( ( 1 - 8 T + 270 T^{2} - 1764 T^{3} + 34598 T^{4} - 1764 p T^{5} + 270 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
97 \( 1 - 316 T^{2} + 65324 T^{4} - 9550388 T^{6} + 1045123990 T^{8} - 9550388 p^{2} T^{10} + 65324 p^{4} T^{12} - 316 p^{6} T^{14} + p^{8} T^{16} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−3.91709286893344988738954271814, −3.89054495126938818068238641342, −3.81991768666813546412031111495, −3.53367275987610326336762585147, −3.42548824719958414197157570818, −3.40998760491534980879845964493, −3.38047141408683041205908817689, −3.21644645913716264930102700117, −2.86860946908592228367776784242, −2.51134881634361145861186343381, −2.50882715273053985334232942987, −2.50832698312760097287876785195, −2.38139494358282878782700996908, −2.21913288680668041513862910580, −2.05634196327415386491053317800, −2.01478749951440108450856737493, −1.82760732286332101745162688280, −1.49331525996767358119624954116, −1.39776402200889714222576843490, −1.09074619162606516455815478407, −1.05653088922196797732773567156, −0.994225928135952564754297496753, −0.53046410276135328497254604951, −0.26186588378419312036299519379, −0.10807177673584569866187233578, 0.10807177673584569866187233578, 0.26186588378419312036299519379, 0.53046410276135328497254604951, 0.994225928135952564754297496753, 1.05653088922196797732773567156, 1.09074619162606516455815478407, 1.39776402200889714222576843490, 1.49331525996767358119624954116, 1.82760732286332101745162688280, 2.01478749951440108450856737493, 2.05634196327415386491053317800, 2.21913288680668041513862910580, 2.38139494358282878782700996908, 2.50832698312760097287876785195, 2.50882715273053985334232942987, 2.51134881634361145861186343381, 2.86860946908592228367776784242, 3.21644645913716264930102700117, 3.38047141408683041205908817689, 3.40998760491534980879845964493, 3.42548824719958414197157570818, 3.53367275987610326336762585147, 3.81991768666813546412031111495, 3.89054495126938818068238641342, 3.91709286893344988738954271814

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

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