Properties

Label 2-1452-1.1-c3-0-31
Degree $2$
Conductor $1452$
Sign $1$
Analytic cond. $85.6707$
Root an. cond. $9.25585$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s + 16.3·5-s + 33.7·7-s + 9·9-s − 53.1·13-s + 48.9·15-s + 86.8·17-s − 154.·19-s + 101.·21-s + 185.·23-s + 140.·25-s + 27·27-s + 115.·29-s − 193.·31-s + 549.·35-s − 123.·37-s − 159.·39-s − 144.·41-s + 317.·43-s + 146.·45-s + 492.·47-s + 794.·49-s + 260.·51-s + 31.4·53-s − 462.·57-s + 202.·59-s + 284.·61-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.45·5-s + 1.82·7-s + 0.333·9-s − 1.13·13-s + 0.841·15-s + 1.23·17-s − 1.86·19-s + 1.05·21-s + 1.68·23-s + 1.12·25-s + 0.192·27-s + 0.739·29-s − 1.12·31-s + 2.65·35-s − 0.550·37-s − 0.654·39-s − 0.548·41-s + 1.12·43-s + 0.486·45-s + 1.52·47-s + 2.31·49-s + 0.715·51-s + 0.0814·53-s − 1.07·57-s + 0.447·59-s + 0.596·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1452 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1452 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1452\)    =    \(2^{2} \cdot 3 \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(85.6707\)
Root analytic conductor: \(9.25585\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1452,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(4.766229274\)
\(L(\frac12)\) \(\approx\) \(4.766229274\)
\(L(\frac{5}{2})\) 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 \( 1 - 3T \)
11 \( 1 \)
good5 \( 1 - 16.3T + 125T^{2} \)
7 \( 1 - 33.7T + 343T^{2} \)
13 \( 1 + 53.1T + 2.19e3T^{2} \)
17 \( 1 - 86.8T + 4.91e3T^{2} \)
19 \( 1 + 154.T + 6.85e3T^{2} \)
23 \( 1 - 185.T + 1.21e4T^{2} \)
29 \( 1 - 115.T + 2.43e4T^{2} \)
31 \( 1 + 193.T + 2.97e4T^{2} \)
37 \( 1 + 123.T + 5.06e4T^{2} \)
41 \( 1 + 144.T + 6.89e4T^{2} \)
43 \( 1 - 317.T + 7.95e4T^{2} \)
47 \( 1 - 492.T + 1.03e5T^{2} \)
53 \( 1 - 31.4T + 1.48e5T^{2} \)
59 \( 1 - 202.T + 2.05e5T^{2} \)
61 \( 1 - 284.T + 2.26e5T^{2} \)
67 \( 1 - 275.T + 3.00e5T^{2} \)
71 \( 1 - 646.T + 3.57e5T^{2} \)
73 \( 1 - 38.8T + 3.89e5T^{2} \)
79 \( 1 + 1.01e3T + 4.93e5T^{2} \)
83 \( 1 + 1.06e3T + 5.71e5T^{2} \)
89 \( 1 + 1.05e3T + 7.04e5T^{2} \)
97 \( 1 - 292.T + 9.12e5T^{2} \)
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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

−9.002432481726407259861059313665, −8.527552819871213353799423729547, −7.57396328800058933581524913658, −6.86681874136931036977918634131, −5.58990501830212038642038071208, −5.10769640926937496049065384384, −4.20207611072484682123459720503, −2.64997696928835569545522501455, −2.01564360986475615084385196481, −1.15465518748266874902934504586, 1.15465518748266874902934504586, 2.01564360986475615084385196481, 2.64997696928835569545522501455, 4.20207611072484682123459720503, 5.10769640926937496049065384384, 5.58990501830212038642038071208, 6.86681874136931036977918634131, 7.57396328800058933581524913658, 8.527552819871213353799423729547, 9.002432481726407259861059313665

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