Properties

Label 2-5472-1.1-c1-0-32
Degree $2$
Conductor $5472$
Sign $1$
Analytic cond. $43.6941$
Root an. cond. $6.61015$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 1.56·5-s + 3·7-s + 3.56·11-s + 2.56·13-s + 8.12·17-s + 19-s − 1.43·23-s − 2.56·25-s + 7.68·29-s + 0.876·31-s − 4.68·35-s − 1.12·37-s − 4·41-s + 9.56·43-s − 8.68·47-s + 2·49-s + 8.56·53-s − 5.56·55-s + 8.56·59-s − 5.80·61-s − 4·65-s − 4.56·67-s − 12.2·71-s + 7.24·73-s + 10.6·77-s + 10·79-s − 7.36·83-s + ⋯
L(s)  = 1  − 0.698·5-s + 1.13·7-s + 1.07·11-s + 0.710·13-s + 1.97·17-s + 0.229·19-s − 0.299·23-s − 0.512·25-s + 1.42·29-s + 0.157·31-s − 0.791·35-s − 0.184·37-s − 0.624·41-s + 1.45·43-s − 1.26·47-s + 0.285·49-s + 1.17·53-s − 0.749·55-s + 1.11·59-s − 0.743·61-s − 0.496·65-s − 0.557·67-s − 1.45·71-s + 0.848·73-s + 1.21·77-s + 1.12·79-s − 0.808·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5472 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5472\)    =    \(2^{5} \cdot 3^{2} \cdot 19\)
Sign: $1$
Analytic conductor: \(43.6941\)
Root analytic conductor: \(6.61015\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5472} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 5472,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.525140013\)
\(L(\frac12)\) \(\approx\) \(2.525140013\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
19 \( 1 - T \)
good5 \( 1 + 1.56T + 5T^{2} \)
7 \( 1 - 3T + 7T^{2} \)
11 \( 1 - 3.56T + 11T^{2} \)
13 \( 1 - 2.56T + 13T^{2} \)
17 \( 1 - 8.12T + 17T^{2} \)
23 \( 1 + 1.43T + 23T^{2} \)
29 \( 1 - 7.68T + 29T^{2} \)
31 \( 1 - 0.876T + 31T^{2} \)
37 \( 1 + 1.12T + 37T^{2} \)
41 \( 1 + 4T + 41T^{2} \)
43 \( 1 - 9.56T + 43T^{2} \)
47 \( 1 + 8.68T + 47T^{2} \)
53 \( 1 - 8.56T + 53T^{2} \)
59 \( 1 - 8.56T + 59T^{2} \)
61 \( 1 + 5.80T + 61T^{2} \)
67 \( 1 + 4.56T + 67T^{2} \)
71 \( 1 + 12.2T + 71T^{2} \)
73 \( 1 - 7.24T + 73T^{2} \)
79 \( 1 - 10T + 79T^{2} \)
83 \( 1 + 7.36T + 83T^{2} \)
89 \( 1 + 9.36T + 89T^{2} \)
97 \( 1 + 1.12T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.172817810352951082318929504919, −7.60684804537190771448330602739, −6.82532182922713467908780470717, −5.94312904475062751970603614611, −5.28049122798695779775084390241, −4.35730989303304622860553620190, −3.80242894078641353783412360705, −2.97834202402391071147252240709, −1.60667607946198771711868199936, −0.961262508287906583155104181122, 0.961262508287906583155104181122, 1.60667607946198771711868199936, 2.97834202402391071147252240709, 3.80242894078641353783412360705, 4.35730989303304622860553620190, 5.28049122798695779775084390241, 5.94312904475062751970603614611, 6.82532182922713467908780470717, 7.60684804537190771448330602739, 8.172817810352951082318929504919

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