Properties

Label 16-1216e8-1.1-c1e8-0-1
Degree $16$
Conductor $4.780\times 10^{24}$
Sign $1$
Analytic cond. $7.90106\times 10^{7}$
Root an. cond. $3.11605$
Motivic weight $1$
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  + 24·9-s − 28·17-s + 18·25-s − 10·49-s + 44·73-s + 324·81-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 672·153-s + 157-s + 163-s + 167-s − 104·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯
L(s)  = 1  + 8·9-s − 6.79·17-s + 18/5·25-s − 1.42·49-s + 5.14·73-s + 36·81-s + 6/11·121-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 − 54.3·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 8·169-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{48} \cdot 19^{8}\)
Sign: $1$
Analytic conductor: \(7.90106\times 10^{7}\)
Root analytic conductor: \(3.11605\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1216} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{48} \cdot 19^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.5788200825\)
\(L(\frac12)\) \(\approx\) \(0.5788200825\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( ( 1 - p T^{2} )^{4} \)
good3 \( ( 1 - p T^{2} )^{8} \)
5 \( ( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} )^{2}( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \)
7 \( ( 1 + 5 T^{2} - 24 T^{4} + 5 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
11 \( ( 1 - 3 T^{2} - 112 T^{4} - 3 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
13 \( ( 1 + p T^{2} )^{8} \)
17 \( ( 1 + 7 T + 32 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{4} \)
23 \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{4} \)
29 \( ( 1 + p T^{2} )^{8} \)
31 \( ( 1 + p T^{2} )^{8} \)
37 \( ( 1 + p T^{2} )^{8} \)
41 \( ( 1 - p T^{2} )^{8} \)
43 \( ( 1 + 85 T^{2} + 5376 T^{4} + 85 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
47 \( ( 1 - 75 T^{2} + 3416 T^{4} - 75 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
53 \( ( 1 + p T^{2} )^{8} \)
59 \( ( 1 - p T^{2} )^{8} \)
61 \( ( 1 - 15 T + 164 T^{2} - 15 p T^{3} + p^{2} T^{4} )^{2}( 1 + 15 T + 164 T^{2} + 15 p T^{3} + p^{2} T^{4} )^{2} \)
67 \( ( 1 - p T^{2} )^{8} \)
71 \( ( 1 + p T^{2} )^{8} \)
73 \( ( 1 - 11 T + 48 T^{2} - 11 p T^{3} + p^{2} T^{4} )^{4} \)
79 \( ( 1 + p T^{2} )^{8} \)
83 \( ( 1 + 90 T^{2} + p^{2} T^{4} )^{4} \)
89 \( ( 1 - p T^{2} )^{8} \)
97 \( ( 1 - p T^{2} )^{8} \)
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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

−4.19340999346431843366418750335, −4.11936313809395067975247173826, −3.81443903747514262171292861300, −3.77620183315409154081257765831, −3.76330034538093440298926728770, −3.61459486615460759189219029421, −3.61235822360394227683251533727, −3.56513336553016587471752218819, −3.12142272396226625215652180262, −2.68079062776598013186740830543, −2.65904015033544259890180611304, −2.61257758447966562836728982884, −2.44661635679820042316832338829, −2.43090014053860003587199355051, −1.96166224219467299219958504918, −1.96041085499069111032198366682, −1.95791676164844489658401001448, −1.82195675813168942972789442970, −1.51094435064327148319910965400, −1.48741051580411768737481763335, −1.16033372492497851735094879424, −0.947081517715189922417658353834, −0.944978688094843935553310537779, −0.793777972022043025629662754543, −0.05137820674468295480397290008, 0.05137820674468295480397290008, 0.793777972022043025629662754543, 0.944978688094843935553310537779, 0.947081517715189922417658353834, 1.16033372492497851735094879424, 1.48741051580411768737481763335, 1.51094435064327148319910965400, 1.82195675813168942972789442970, 1.95791676164844489658401001448, 1.96041085499069111032198366682, 1.96166224219467299219958504918, 2.43090014053860003587199355051, 2.44661635679820042316832338829, 2.61257758447966562836728982884, 2.65904015033544259890180611304, 2.68079062776598013186740830543, 3.12142272396226625215652180262, 3.56513336553016587471752218819, 3.61235822360394227683251533727, 3.61459486615460759189219029421, 3.76330034538093440298926728770, 3.77620183315409154081257765831, 3.81443903747514262171292861300, 4.11936313809395067975247173826, 4.19340999346431843366418750335

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

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