Properties

Label 32-772e16-1.1-c0e16-0-0
Degree $32$
Conductor $1.592\times 10^{46}$
Sign $1$
Analytic cond. $2.35716\times 10^{-7}$
Root an. cond. $0.620707$
Motivic weight $0$
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  − 8·17-s + 8·49-s − 8·81-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4·169-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 + 239-s + 241-s + ⋯
L(s)  = 1  − 8·17-s + 8·49-s − 8·81-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4·169-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 + 239-s + 241-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 193^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 193^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(2^{32} \cdot 193^{16}\)
Sign: $1$
Analytic conductor: \(2.35716\times 10^{-7}\)
Root analytic conductor: \(0.620707\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{772} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 2^{32} \cdot 193^{16} ,\ ( \ : [0]^{16} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1604233255\)
\(L(\frac12)\) \(\approx\) \(0.1604233255\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T^{8} + T^{16} \)
193 \( ( 1 + T^{4} )^{4} \)
good3 \( ( 1 + T^{4} )^{8} \)
5 \( ( 1 - T^{4} + T^{8} )^{2}( 1 - T^{8} + T^{16} ) \)
7 \( ( 1 - T^{2} + T^{4} )^{8} \)
11 \( ( 1 + T^{16} )^{2} \)
13 \( ( 1 - T^{2} + T^{4} )^{4}( 1 - T^{8} + T^{16} ) \)
17 \( ( 1 + T + T^{2} )^{8}( 1 - T^{8} + T^{16} ) \)
19 \( 1 - T^{16} + T^{32} \)
23 \( ( 1 + T^{8} )^{4} \)
29 \( ( 1 - T^{2} + T^{4} )^{4}( 1 - T^{8} + T^{16} ) \)
31 \( ( 1 - T^{8} + T^{16} )^{2} \)
37 \( ( 1 - T^{4} + T^{8} )^{2}( 1 + T^{8} )^{2} \)
41 \( ( 1 + T^{4} )^{4}( 1 - T^{8} + T^{16} ) \)
43 \( ( 1 + T^{2} )^{16} \)
47 \( 1 - T^{16} + T^{32} \)
53 \( ( 1 + T^{4} )^{4}( 1 - T^{8} + T^{16} ) \)
59 \( ( 1 - T^{4} + T^{8} )^{4} \)
61 \( ( 1 - T^{4} + T^{8} )^{2}( 1 + T^{8} )^{2} \)
67 \( ( 1 + T^{8} )^{4} \)
71 \( ( 1 + T^{16} )^{2} \)
73 \( ( 1 + T^{4} )^{4}( 1 - T^{8} + T^{16} ) \)
79 \( 1 - T^{16} + T^{32} \)
83 \( ( 1 - T^{8} + T^{16} )^{2} \)
89 \( ( 1 + T^{2} )^{8}( 1 + T^{8} )^{2} \)
97 \( ( 1 + T^{8} )^{2}( 1 - T^{8} + T^{16} ) \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−2.90060114328725907837314534759, −2.81878817228431584719077475117, −2.79033598182948095848059520565, −2.74704202351608846512647275955, −2.70454663379331532259835206004, −2.70431095472434333458226947391, −2.69835023649434470594164621772, −2.40991930193802078733944442266, −2.40978064899841668991239670395, −2.30372572472568799329027068713, −2.19888845135035083034594986170, −2.17804168817621273354707273559, −1.90924306303309231507689577665, −1.89005802164656225508777371854, −1.84974517055905239695531485583, −1.79641561234089062297370619511, −1.77458625032616306547956977416, −1.69640654096583928862495610121, −1.68730861585721120479540025380, −1.30861616326027077854526527936, −1.14505044608192245189447664147, −1.10002274406697028666163995662, −0.864641721165017708832117213964, −0.792217138681178982106963346258, −0.55502463297254491571726824398, 0.55502463297254491571726824398, 0.792217138681178982106963346258, 0.864641721165017708832117213964, 1.10002274406697028666163995662, 1.14505044608192245189447664147, 1.30861616326027077854526527936, 1.68730861585721120479540025380, 1.69640654096583928862495610121, 1.77458625032616306547956977416, 1.79641561234089062297370619511, 1.84974517055905239695531485583, 1.89005802164656225508777371854, 1.90924306303309231507689577665, 2.17804168817621273354707273559, 2.19888845135035083034594986170, 2.30372572472568799329027068713, 2.40978064899841668991239670395, 2.40991930193802078733944442266, 2.69835023649434470594164621772, 2.70431095472434333458226947391, 2.70454663379331532259835206004, 2.74704202351608846512647275955, 2.79033598182948095848059520565, 2.81878817228431584719077475117, 2.90060114328725907837314534759

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

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