Properties

Label 32-765e16-1.1-c0e16-0-0
Degree $32$
Conductor $1.376\times 10^{46}$
Sign $1$
Analytic cond. $2.03749\times 10^{-7}$
Root an. cond. $0.617887$
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  + 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 + 251-s + 257-s + 263-s + 269-s + ⋯
L(s)  = 1  + 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 + 251-s + 257-s + 263-s + 269-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 5^{16} \cdot 17^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 5^{16} \cdot 17^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(3^{32} \cdot 5^{16} \cdot 17^{16}\)
Sign: $1$
Analytic conductor: \(2.03749\times 10^{-7}\)
Root analytic conductor: \(0.617887\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{765} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 3^{32} \cdot 5^{16} \cdot 17^{16} ,\ ( \ : [0]^{16} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2821188653\)
\(L(\frac12)\) \(\approx\) \(0.2821188653\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−2.97730045998207573032930133339, −2.94710363602184265221011741971, −2.84930623426575695339333047017, −2.82012841196487127354143360108, −2.81836618054093303435389349861, −2.72097161958540813175416436488, −2.68605633806220370917440214395, −2.38917756139351211811678578121, −2.28497297713887498148155707920, −2.28369476668750028460570189517, −2.18680474008867270906849577472, −2.12610440049152456759970863715, −2.05337541996567255759698809051, −2.01592566589998546916894904115, −1.79538733170427666930922500126, −1.70468501177738606497302629244, −1.64524018631882813122119543710, −1.61598307854116221324870686483, −1.55814090834279787235449273614, −1.29638118279247382254072612825, −1.25131028205294972961245863721, −1.06554931058569505073489949014, −0.862381926309192441169465725467, −0.829679063317857432635731126611, −0.816155180189329016130218229162, 0.816155180189329016130218229162, 0.829679063317857432635731126611, 0.862381926309192441169465725467, 1.06554931058569505073489949014, 1.25131028205294972961245863721, 1.29638118279247382254072612825, 1.55814090834279787235449273614, 1.61598307854116221324870686483, 1.64524018631882813122119543710, 1.70468501177738606497302629244, 1.79538733170427666930922500126, 2.01592566589998546916894904115, 2.05337541996567255759698809051, 2.12610440049152456759970863715, 2.18680474008867270906849577472, 2.28369476668750028460570189517, 2.28497297713887498148155707920, 2.38917756139351211811678578121, 2.68605633806220370917440214395, 2.72097161958540813175416436488, 2.81836618054093303435389349861, 2.82012841196487127354143360108, 2.84930623426575695339333047017, 2.94710363602184265221011741971, 2.97730045998207573032930133339

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

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