Properties

Label 2-5472-1.1-c1-0-31
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.56·5-s + 3·7-s − 0.561·11-s − 1.56·13-s − 0.123·17-s + 19-s − 5.56·23-s + 1.56·25-s − 4.68·29-s + 9.12·31-s + 7.68·35-s + 7.12·37-s − 4·41-s + 5.43·43-s + 3.68·47-s + 2·49-s + 4.43·53-s − 1.43·55-s + 4.43·59-s + 14.8·61-s − 4·65-s − 0.438·67-s + 4.24·71-s − 9.24·73-s − 1.68·77-s + 10·79-s + 17.3·83-s + ⋯
L(s)  = 1  + 1.14·5-s + 1.13·7-s − 0.169·11-s − 0.433·13-s − 0.0298·17-s + 0.229·19-s − 1.15·23-s + 0.312·25-s − 0.869·29-s + 1.63·31-s + 1.29·35-s + 1.17·37-s − 0.624·41-s + 0.829·43-s + 0.537·47-s + 0.285·49-s + 0.609·53-s − 0.193·55-s + 0.577·59-s + 1.89·61-s − 0.496·65-s − 0.0535·67-s + 0.503·71-s − 1.08·73-s − 0.191·77-s + 1.12·79-s + 1.90·83-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

\(L(1)\) \(\approx\) \(2.888061977\)
\(L(\frac12)\) \(\approx\) \(2.888061977\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.56T + 5T^{2} \)
7 \( 1 - 3T + 7T^{2} \)
11 \( 1 + 0.561T + 11T^{2} \)
13 \( 1 + 1.56T + 13T^{2} \)
17 \( 1 + 0.123T + 17T^{2} \)
23 \( 1 + 5.56T + 23T^{2} \)
29 \( 1 + 4.68T + 29T^{2} \)
31 \( 1 - 9.12T + 31T^{2} \)
37 \( 1 - 7.12T + 37T^{2} \)
41 \( 1 + 4T + 41T^{2} \)
43 \( 1 - 5.43T + 43T^{2} \)
47 \( 1 - 3.68T + 47T^{2} \)
53 \( 1 - 4.43T + 53T^{2} \)
59 \( 1 - 4.43T + 59T^{2} \)
61 \( 1 - 14.8T + 61T^{2} \)
67 \( 1 + 0.438T + 67T^{2} \)
71 \( 1 - 4.24T + 71T^{2} \)
73 \( 1 + 9.24T + 73T^{2} \)
79 \( 1 - 10T + 79T^{2} \)
83 \( 1 - 17.3T + 83T^{2} \)
89 \( 1 - 15.3T + 89T^{2} \)
97 \( 1 - 7.12T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.042375524290343568199281642834, −7.64163850794416004344796674669, −6.62081544760676749321757023340, −5.93744203980919319427482091975, −5.30073930920283809331793726000, −4.64820783072911342871159965307, −3.77855011782730182528548303337, −2.44895219823249263787187684678, −2.04333652287247103430469472446, −0.942066293094218888838118614618, 0.942066293094218888838118614618, 2.04333652287247103430469472446, 2.44895219823249263787187684678, 3.77855011782730182528548303337, 4.64820783072911342871159965307, 5.30073930920283809331793726000, 5.93744203980919319427482091975, 6.62081544760676749321757023340, 7.64163850794416004344796674669, 8.042375524290343568199281642834

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