Properties

Label 32-3920e16-1.1-c0e16-0-0
Degree $32$
Conductor $3.109\times 10^{57}$
Sign $1$
Analytic cond. $46035.3$
Root an. cond. $1.39869$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 8·53-s + 4·81-s + 16·113-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  + 8·53-s + 4·81-s + 16·113-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

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 5^{16} \cdot 7^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 5^{16} \cdot 7^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(2^{64} \cdot 5^{16} \cdot 7^{32}\)
Sign: $1$
Analytic conductor: \(46035.3\)
Root analytic conductor: \(1.39869\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3920} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 2^{64} \cdot 5^{16} \cdot 7^{32} ,\ ( \ : [0]^{16} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.754260239\)
\(L(\frac12)\) \(\approx\) \(2.754260239\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−2.28029721270195856769038844325, −2.22473271480746796021135779839, −2.21552425600660151434293592183, −2.19674684513873403396136706128, −2.11563345474380374881956938979, −2.11460060210026892770636273291, −1.90464267110288267970900851895, −1.85631519466179718673316781499, −1.80036553905066311931658170668, −1.65924969479245898213628193011, −1.41574069744748303335841413108, −1.39682455406892899880959016098, −1.35239884219236439702332284446, −1.34256831903997319395807278827, −1.33863972696534547003139124745, −1.28301217238724581554223030146, −1.23375566306066413544447781518, −1.04984633787087071444886035687, −0.869743149276443668558106479813, −0.822077296150696401396145358258, −0.76041929258738686985282340500, −0.68917433027391272190956518837, −0.68871297106483726106070601002, −0.51374411589060138566693037368, −0.21094263446537228234223033387, 0.21094263446537228234223033387, 0.51374411589060138566693037368, 0.68871297106483726106070601002, 0.68917433027391272190956518837, 0.76041929258738686985282340500, 0.822077296150696401396145358258, 0.869743149276443668558106479813, 1.04984633787087071444886035687, 1.23375566306066413544447781518, 1.28301217238724581554223030146, 1.33863972696534547003139124745, 1.34256831903997319395807278827, 1.35239884219236439702332284446, 1.39682455406892899880959016098, 1.41574069744748303335841413108, 1.65924969479245898213628193011, 1.80036553905066311931658170668, 1.85631519466179718673316781499, 1.90464267110288267970900851895, 2.11460060210026892770636273291, 2.11563345474380374881956938979, 2.19674684513873403396136706128, 2.21552425600660151434293592183, 2.22473271480746796021135779839, 2.28029721270195856769038844325

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

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