Properties

Label 16-40e16-1.1-c1e8-0-5
Degree $16$
Conductor $4.295\times 10^{25}$
Sign $1$
Analytic cond. $7.09866\times 10^{8}$
Root an. cond. $3.57436$
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  − 32·41-s − 48·49-s − 36·81-s + 16·89-s + 72·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 24·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯
L(s)  = 1  − 4.99·41-s − 6.85·49-s − 4·81-s + 1.69·89-s + 6.54·121-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 − 1.84·169-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 + 0.0663·227-s + 0.0660·229-s + 0.0655·233-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{48} \cdot 5^{16}\)
Sign: $1$
Analytic conductor: \(7.09866\times 10^{8}\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{48} \cdot 5^{16} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.504505917\)
\(L(\frac12)\) \(\approx\) \(1.504505917\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good3 \( ( 1 + p^{2} T^{4} )^{4} \)
7 \( ( 1 + 12 T^{2} + p^{2} T^{4} )^{4} \)
11 \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{4} \)
13 \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{4} \)
17 \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{4} \)
19 \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{4} \)
23 \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{4} \)
29 \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{4} \)
31 \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{4} \)
37 \( ( 1 - 66 T^{2} + p^{2} T^{4} )^{4} \)
41 \( ( 1 + 4 T + p T^{2} )^{8} \)
43 \( ( 1 - 80 T^{2} + p^{2} T^{4} )^{4} \)
47 \( ( 1 + 76 T^{2} + p^{2} T^{4} )^{4} \)
53 \( ( 1 - p T^{2} )^{8} \)
59 \( ( 1 - 114 T^{2} + p^{2} T^{4} )^{4} \)
61 \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{4} \)
67 \( ( 1 - 128 T^{2} + p^{2} T^{4} )^{4} \)
71 \( ( 1 + 94 T^{2} + p^{2} T^{4} )^{4} \)
73 \( ( 1 + 122 T^{2} + p^{2} T^{4} )^{4} \)
79 \( ( 1 + 110 T^{2} + p^{2} T^{4} )^{4} \)
83 \( ( 1 - 16 T^{2} + p^{2} T^{4} )^{4} \)
89 \( ( 1 - 2 T + p T^{2} )^{8} \)
97 \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{4} \)
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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

−3.82735242538910217186348391247, −3.79557429084374096162018441173, −3.58697076141385145231122429307, −3.51199236891614315933432793329, −3.46705258049797559335145238696, −3.40934392789887687042232423191, −3.31083414108393185761599492123, −3.09321895281421048473568916542, −3.00343674054883733757813618222, −2.83750464082889698456162455589, −2.69720058437501090112049134249, −2.67627034417922973715446433289, −2.40980621806028837572283963758, −2.23555568025589576347109645376, −1.90908187040050605445836791699, −1.83793835563667416730803563954, −1.76453684665982729058721692108, −1.67115220731945151723606599809, −1.57983943697422879962654576470, −1.46046292971988945974909376639, −1.14865351079801955579773138245, −0.957599599804535232267289757337, −0.53112305967938897472796508173, −0.33425336309168928140129925263, −0.19257897509366683125021025519, 0.19257897509366683125021025519, 0.33425336309168928140129925263, 0.53112305967938897472796508173, 0.957599599804535232267289757337, 1.14865351079801955579773138245, 1.46046292971988945974909376639, 1.57983943697422879962654576470, 1.67115220731945151723606599809, 1.76453684665982729058721692108, 1.83793835563667416730803563954, 1.90908187040050605445836791699, 2.23555568025589576347109645376, 2.40980621806028837572283963758, 2.67627034417922973715446433289, 2.69720058437501090112049134249, 2.83750464082889698456162455589, 3.00343674054883733757813618222, 3.09321895281421048473568916542, 3.31083414108393185761599492123, 3.40934392789887687042232423191, 3.46705258049797559335145238696, 3.51199236891614315933432793329, 3.58697076141385145231122429307, 3.79557429084374096162018441173, 3.82735242538910217186348391247

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

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