Properties

Label 16-96e16-1.1-c1e8-0-0
Degree $16$
Conductor $5.204\times 10^{31}$
Sign $1$
Analytic cond. $8.60115\times 10^{14}$
Root an. cond. $8.57846$
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  + 8·7-s − 16·25-s + 24·31-s + 8·49-s + 16·73-s + 56·79-s + 72·103-s − 40·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 48·169-s + 173-s − 128·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  + 3.02·7-s − 3.19·25-s + 4.31·31-s + 8/7·49-s + 1.87·73-s + 6.30·79-s + 7.09·103-s − 3.63·121-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 − 3.69·169-s + 0.0760·173-s − 9.67·175-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{80} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{80} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{80} \cdot 3^{16}\)
Sign: $1$
Analytic conductor: \(8.60115\times 10^{14}\)
Root analytic conductor: \(8.57846\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{80} \cdot 3^{16} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(27.61855317\)
\(L(\frac12)\) \(\approx\) \(27.61855317\)
\(L(\frac{3}{2})\) 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 \( ( 1 + 8 T^{2} + 38 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
7 \( ( 1 - 2 T + 8 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{4} \)
11 \( ( 1 + 20 T^{2} + 230 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
13 \( ( 1 + 12 T^{2} + p^{2} T^{4} )^{4} \)
17 \( ( 1 + 28 T^{2} + 662 T^{4} + 28 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
19 \( ( 1 + 44 T^{2} + 1094 T^{4} + 44 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
23 \( ( 1 + 12 T^{2} + 646 T^{4} + 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
29 \( ( 1 + 40 T^{2} + 710 T^{4} + 40 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
31 \( ( 1 - 6 T + 64 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{4} \)
37 \( ( 1 + 104 T^{2} + 4994 T^{4} + 104 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
41 \( ( 1 + 60 T^{2} + 4150 T^{4} + 60 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
43 \( ( 1 + 108 T^{2} + 5606 T^{4} + 108 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
47 \( ( 1 - 20 T^{2} + 4070 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
53 \( ( 1 + 104 T^{2} + 8294 T^{4} + 104 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
59 \( ( 1 + 76 T^{2} + 6614 T^{4} + 76 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
61 \( ( 1 + 200 T^{2} + 16994 T^{4} + 200 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
67 \( ( 1 + 102 T^{2} + p^{2} T^{4} )^{4} \)
71 \( ( 1 + 92 T^{2} + 5030 T^{4} + 92 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
73 \( ( 1 - 4 T + 122 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{4} \)
79 \( ( 1 - 14 T + 200 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{4} \)
83 \( ( 1 + 308 T^{2} + 37382 T^{4} + 308 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
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

−3.29606038965032503696508916008, −2.92191405297734064502918328095, −2.80935816076199211318033819870, −2.69971533691980603114463741435, −2.63367790626270340584778924986, −2.53475854125810454758784571120, −2.46239677295517796245838055217, −2.40120810211434111747445634559, −2.19401052241475051875409903979, −2.06408580441076774487108654858, −1.89137154322445915183308966480, −1.82284953226461367609744149339, −1.77686372583844956941452355118, −1.69793796335124873103427171773, −1.66678081978294004542760807315, −1.65525907361738288423917510910, −1.49135887340370661996530356035, −1.00803416753245694295410635162, −1.00026816216361051464027449027, −0.78203120092131269924196104676, −0.76741747547284946892708218557, −0.72910350531520943437754946736, −0.59306958411625150675633905494, −0.45231703508638061521685891002, −0.15384538730485727007168756666, 0.15384538730485727007168756666, 0.45231703508638061521685891002, 0.59306958411625150675633905494, 0.72910350531520943437754946736, 0.76741747547284946892708218557, 0.78203120092131269924196104676, 1.00026816216361051464027449027, 1.00803416753245694295410635162, 1.49135887340370661996530356035, 1.65525907361738288423917510910, 1.66678081978294004542760807315, 1.69793796335124873103427171773, 1.77686372583844956941452355118, 1.82284953226461367609744149339, 1.89137154322445915183308966480, 2.06408580441076774487108654858, 2.19401052241475051875409903979, 2.40120810211434111747445634559, 2.46239677295517796245838055217, 2.53475854125810454758784571120, 2.63367790626270340584778924986, 2.69971533691980603114463741435, 2.80935816076199211318033819870, 2.92191405297734064502918328095, 3.29606038965032503696508916008

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

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