Properties

Label 16-3525e8-1.1-c0e8-0-0
Degree $16$
Conductor $2.384\times 10^{28}$
Sign $1$
Analytic cond. $91.7339$
Root an. cond. $1.32634$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·16-s − 16·61-s + 81-s − 8·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-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 + 239-s + 241-s + ⋯
L(s)  = 1  + 2·16-s − 16·61-s + 81-s − 8·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-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 + 239-s + 241-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{16} \cdot 47^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{16} \cdot 47^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(3^{8} \cdot 5^{16} \cdot 47^{8}\)
Sign: $1$
Analytic conductor: \(91.7339\)
Root analytic conductor: \(1.32634\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3525} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 3^{8} \cdot 5^{16} \cdot 47^{8} ,\ ( \ : [0]^{8} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.009719858733\)
\(L(\frac12)\) \(\approx\) \(0.009719858733\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T^{4} + T^{8} \)
5 \( 1 \)
47 \( ( 1 + T^{4} )^{2} \)
good2 \( ( 1 - T^{4} + T^{8} )^{2} \)
7 \( ( 1 - T^{4} + T^{8} )^{2} \)
11 \( ( 1 + T^{2} )^{8} \)
13 \( ( 1 + T^{4} )^{4} \)
17 \( ( 1 - T^{4} + T^{8} )^{2} \)
19 \( ( 1 + T^{2} )^{8} \)
23 \( ( 1 + T^{4} )^{4} \)
29 \( ( 1 - T )^{8}( 1 + T )^{8} \)
31 \( ( 1 - T )^{8}( 1 + T )^{8} \)
37 \( ( 1 + T^{4} )^{4} \)
41 \( ( 1 + T^{2} )^{8} \)
43 \( ( 1 + T^{4} )^{4} \)
53 \( ( 1 + T^{4} )^{4} \)
59 \( ( 1 - T^{2} + T^{4} )^{4} \)
61 \( ( 1 + T )^{16} \)
67 \( ( 1 + T^{4} )^{4} \)
71 \( ( 1 - T + T^{2} )^{4}( 1 + T + T^{2} )^{4} \)
73 \( ( 1 + T^{4} )^{4} \)
79 \( ( 1 + T^{2} )^{8} \)
83 \( ( 1 - T^{4} + T^{8} )^{2} \)
89 \( ( 1 + T^{2} )^{8} \)
97 \( ( 1 + T^{4} )^{4} \)
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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

−3.69797371147580855465184858353, −3.49797638240591585257606217050, −3.46383268316202455587347645985, −3.35122049055856724677830405214, −3.33003097663824771469702310679, −3.27549092526278840409852677593, −3.16557661153671809899149674169, −2.81497178077671507207623184998, −2.80023084825287963260119746084, −2.75964135765979437261334562891, −2.67330172861988962525983764887, −2.59494605629484204668092222342, −2.39098088347428802699364599651, −2.25806995042473473997359765252, −2.01320590940827448571370620370, −1.72309836787225465382228339224, −1.69510109899488286754149811894, −1.65463242015511371176126713542, −1.47170732745614127750193532824, −1.38117590986210071406816170902, −1.26335389290067736961625803499, −1.11986188032445422267859595061, −1.10574205523218998757335827193, −0.52990121037317696304302490013, −0.02545428596973558925332537796, 0.02545428596973558925332537796, 0.52990121037317696304302490013, 1.10574205523218998757335827193, 1.11986188032445422267859595061, 1.26335389290067736961625803499, 1.38117590986210071406816170902, 1.47170732745614127750193532824, 1.65463242015511371176126713542, 1.69510109899488286754149811894, 1.72309836787225465382228339224, 2.01320590940827448571370620370, 2.25806995042473473997359765252, 2.39098088347428802699364599651, 2.59494605629484204668092222342, 2.67330172861988962525983764887, 2.75964135765979437261334562891, 2.80023084825287963260119746084, 2.81497178077671507207623184998, 3.16557661153671809899149674169, 3.27549092526278840409852677593, 3.33003097663824771469702310679, 3.35122049055856724677830405214, 3.46383268316202455587347645985, 3.49797638240591585257606217050, 3.69797371147580855465184858353

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

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