Properties

Label 32-799e16-1.1-c0e16-0-0
Degree $32$
Conductor $2.759\times 10^{46}$
Sign $1$
Analytic cond. $4.08566\times 10^{-7}$
Root an. cond. $0.631468$
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  − 4·3-s + 6·9-s + 2·16-s + 4·17-s − 4·27-s − 8·48-s + 2·49-s − 16·51-s + 4·53-s + 4·61-s − 4·71-s − 4·79-s + 2·81-s − 16·83-s + 127-s + 131-s + 137-s + 139-s + 12·144-s − 8·147-s + 149-s + 151-s + 24·153-s + 157-s − 16·159-s + 163-s + 167-s + ⋯
L(s)  = 1  − 4·3-s + 6·9-s + 2·16-s + 4·17-s − 4·27-s − 8·48-s + 2·49-s − 16·51-s + 4·53-s + 4·61-s − 4·71-s − 4·79-s + 2·81-s − 16·83-s + 127-s + 131-s + 137-s + 139-s + 12·144-s − 8·147-s + 149-s + 151-s + 24·153-s + 157-s − 16·159-s + 163-s + 167-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(17^{16} \cdot 47^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(17^{16} \cdot 47^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(17^{16} \cdot 47^{16}\)
Sign: $1$
Analytic conductor: \(4.08566\times 10^{-7}\)
Root analytic conductor: \(0.631468\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{799} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 17^{16} \cdot 47^{16} ,\ ( \ : [0]^{16} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03364897197\)
\(L(\frac12)\) \(\approx\) \(0.03364897197\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4} \)
47 \( ( 1 + T^{2} )^{8} \)
good2 \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \)
3 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{4}( 1 - T^{4} + T^{8} - T^{12} + T^{16} ) \)
5 \( ( 1 + T^{8} )^{4} \)
7 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}( 1 - T^{4} + T^{8} - T^{12} + T^{16} ) \)
11 \( ( 1 + T^{8} )^{4} \)
13 \( ( 1 + T^{2} )^{16} \)
19 \( ( 1 + T^{4} )^{8} \)
23 \( ( 1 + T^{8} )^{4} \)
29 \( ( 1 + T^{8} )^{4} \)
31 \( ( 1 + T^{8} )^{4} \)
37 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}( 1 - T^{4} + T^{8} - T^{12} + T^{16} ) \)
41 \( ( 1 + T^{8} )^{4} \)
43 \( ( 1 + T^{4} )^{8} \)
53 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
59 \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \)
61 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 - T^{4} + T^{8} - T^{12} + T^{16} ) \)
67 \( ( 1 - T )^{16}( 1 + T )^{16} \)
71 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{4}( 1 - T^{4} + T^{8} - T^{12} + T^{16} ) \)
73 \( ( 1 + T^{8} )^{4} \)
79 \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{4}( 1 - T^{4} + T^{8} - T^{12} + T^{16} ) \)
83 \( ( 1 + T )^{16}( 1 + T^{2} )^{8} \)
89 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
97 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2}( 1 - T^{4} + T^{8} - T^{12} + T^{16} ) \)
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

−3.03746057365108343043860821507, −2.88335228674321957833288608823, −2.79135870140498464984674385396, −2.78373412992047753801095905578, −2.77320082433711147667319089125, −2.76180661304722172829474014509, −2.46756349797084567514259080872, −2.43978365373120494013713083807, −2.40324343431032421921264042041, −2.32389001212998377344039451891, −2.15295545342666336722301994920, −2.09975073747178200638069797952, −2.03377961245795082116856778826, −1.90185074092271568990911635741, −1.55137492389192093949885263424, −1.49259719282995548490097771660, −1.48238663923497214537060454852, −1.47059383884631236045086779005, −1.33350149001272049216378001712, −1.22427847911943091610002951886, −1.12032558219798655196954206314, −1.09419371937868918424388460561, −0.983077308087567981076856614977, −0.894786694864230422643487387847, −0.33836518353393647209040843793, 0.33836518353393647209040843793, 0.894786694864230422643487387847, 0.983077308087567981076856614977, 1.09419371937868918424388460561, 1.12032558219798655196954206314, 1.22427847911943091610002951886, 1.33350149001272049216378001712, 1.47059383884631236045086779005, 1.48238663923497214537060454852, 1.49259719282995548490097771660, 1.55137492389192093949885263424, 1.90185074092271568990911635741, 2.03377961245795082116856778826, 2.09975073747178200638069797952, 2.15295545342666336722301994920, 2.32389001212998377344039451891, 2.40324343431032421921264042041, 2.43978365373120494013713083807, 2.46756349797084567514259080872, 2.76180661304722172829474014509, 2.77320082433711147667319089125, 2.78373412992047753801095905578, 2.79135870140498464984674385396, 2.88335228674321957833288608823, 3.03746057365108343043860821507

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

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