Properties

Label 2-39e2-1.1-c3-0-181
Degree $2$
Conductor $1521$
Sign $-1$
Analytic cond. $89.7419$
Root an. cond. $9.47322$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.56·2-s + 12.8·4-s − 2.80·5-s + 9.56·7-s + 21.9·8-s − 12.8·10-s − 39.4·11-s + 43.6·14-s − 2.42·16-s − 2.01·17-s − 60.1·19-s − 35.9·20-s − 179.·22-s − 4.46·23-s − 117.·25-s + 122.·28-s − 140.·29-s + 136.·31-s − 186.·32-s − 9.19·34-s − 26.8·35-s − 185.·37-s − 274.·38-s − 61.5·40-s − 310.·41-s + 427.·43-s − 504.·44-s + ⋯
L(s)  = 1  + 1.61·2-s + 1.60·4-s − 0.251·5-s + 0.516·7-s + 0.969·8-s − 0.405·10-s − 1.08·11-s + 0.832·14-s − 0.0378·16-s − 0.0287·17-s − 0.726·19-s − 0.402·20-s − 1.74·22-s − 0.0405·23-s − 0.936·25-s + 0.826·28-s − 0.900·29-s + 0.788·31-s − 1.03·32-s − 0.0463·34-s − 0.129·35-s − 0.825·37-s − 1.17·38-s − 0.243·40-s − 1.18·41-s + 1.51·43-s − 1.73·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1521\)    =    \(3^{2} \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(89.7419\)
Root analytic conductor: \(9.47322\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1521,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
good2 \( 1 - 4.56T + 8T^{2} \)
5 \( 1 + 2.80T + 125T^{2} \)
7 \( 1 - 9.56T + 343T^{2} \)
11 \( 1 + 39.4T + 1.33e3T^{2} \)
17 \( 1 + 2.01T + 4.91e3T^{2} \)
19 \( 1 + 60.1T + 6.85e3T^{2} \)
23 \( 1 + 4.46T + 1.21e4T^{2} \)
29 \( 1 + 140.T + 2.43e4T^{2} \)
31 \( 1 - 136.T + 2.97e4T^{2} \)
37 \( 1 + 185.T + 5.06e4T^{2} \)
41 \( 1 + 310.T + 6.89e4T^{2} \)
43 \( 1 - 427.T + 7.95e4T^{2} \)
47 \( 1 - 258.T + 1.03e5T^{2} \)
53 \( 1 + 612.T + 1.48e5T^{2} \)
59 \( 1 - 517.T + 2.05e5T^{2} \)
61 \( 1 + 161.T + 2.26e5T^{2} \)
67 \( 1 + 49.8T + 3.00e5T^{2} \)
71 \( 1 + 279.T + 3.57e5T^{2} \)
73 \( 1 - 467.T + 3.89e5T^{2} \)
79 \( 1 - 37.5T + 4.93e5T^{2} \)
83 \( 1 - 76.1T + 5.71e5T^{2} \)
89 \( 1 + 202.T + 7.04e5T^{2} \)
97 \( 1 + 1.17e3T + 9.12e5T^{2} \)
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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

−8.452744697228628296318353397802, −7.73512026242645426103144069223, −6.87476277329832332185646800392, −5.93536270318192295273082171192, −5.26506729296925147544531792352, −4.51205546209074883089296415426, −3.73917704178898812906298892589, −2.73837992008216085097419852136, −1.85421647489021626762030370602, 0, 1.85421647489021626762030370602, 2.73837992008216085097419852136, 3.73917704178898812906298892589, 4.51205546209074883089296415426, 5.26506729296925147544531792352, 5.93536270318192295273082171192, 6.87476277329832332185646800392, 7.73512026242645426103144069223, 8.452744697228628296318353397802

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