Properties

Label 32-975e16-1.1-c0e16-0-0
Degree $32$
Conductor $6.669\times 10^{47}$
Sign $1$
Analytic cond. $9.87615\times 10^{-6}$
Root an. cond. $0.697558$
Motivic weight $0$
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  + 2·9-s − 3·16-s − 16·49-s + 81-s + 127-s + 131-s + 137-s + 139-s − 6·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-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  + 2·9-s − 3·16-s − 16·49-s + 81-s + 127-s + 131-s + 137-s + 139-s − 6·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{32} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{32} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(3^{16} \cdot 5^{32} \cdot 13^{16}\)
Sign: $1$
Analytic conductor: \(9.87615\times 10^{-6}\)
Root analytic conductor: \(0.697558\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{975} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 3^{16} \cdot 5^{32} \cdot 13^{16} ,\ ( \ : [0]^{16} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1791026804\)
\(L(\frac12)\) \(\approx\) \(0.1791026804\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
5 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \)
13 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
good2 \( ( 1 + T^{4} )^{4}( 1 - T^{4} + T^{8} - T^{12} + T^{16} ) \)
7 \( ( 1 + T^{2} )^{16} \)
11 \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \)
17 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \)
19 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
23 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \)
29 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
31 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
37 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \)
41 \( ( 1 + T^{4} )^{4}( 1 - T^{4} + T^{8} - T^{12} + T^{16} ) \)
43 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
47 \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \)
53 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \)
59 \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \)
61 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \)
67 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \)
71 \( ( 1 + T^{4} )^{4}( 1 - T^{4} + T^{8} - T^{12} + T^{16} ) \)
73 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \)
79 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \)
83 \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \)
89 \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \)
97 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−2.93410506087550591529107919330, −2.80570141894750761611355268717, −2.79402913931028338984607056917, −2.71850247024525594941647829745, −2.52404999577901244857108114913, −2.48330500231968448914534153808, −2.38688002882939121398753330893, −2.38303108319699683369234084055, −2.25822554770458533678005329018, −2.13618708428655400869897223792, −2.11381841264904988706063107312, −2.01884709036703632792668304713, −1.77601975087041888877434968187, −1.64223146874534932071114887388, −1.64097405616498299695464749173, −1.63878182803389904539012147292, −1.59755820807059330542658171969, −1.56530442872749372626114935026, −1.52527424760014020740837703021, −1.32959692551193285030458486579, −1.27454563877749017762600835090, −1.10713423397349020545853462097, −1.06072389909271935230527390643, −0.55687960003100934479902586000, −0.35261130869510859212234362975, 0.35261130869510859212234362975, 0.55687960003100934479902586000, 1.06072389909271935230527390643, 1.10713423397349020545853462097, 1.27454563877749017762600835090, 1.32959692551193285030458486579, 1.52527424760014020740837703021, 1.56530442872749372626114935026, 1.59755820807059330542658171969, 1.63878182803389904539012147292, 1.64097405616498299695464749173, 1.64223146874534932071114887388, 1.77601975087041888877434968187, 2.01884709036703632792668304713, 2.11381841264904988706063107312, 2.13618708428655400869897223792, 2.25822554770458533678005329018, 2.38303108319699683369234084055, 2.38688002882939121398753330893, 2.48330500231968448914534153808, 2.52404999577901244857108114913, 2.71850247024525594941647829745, 2.79402913931028338984607056917, 2.80570141894750761611355268717, 2.93410506087550591529107919330

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

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