Properties

Label 2-39e2-1.1-c3-0-20
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.03·2-s + 8.24·4-s − 8.08·5-s + 5.95·7-s − 0.978·8-s + 32.5·10-s − 17.2·11-s − 23.9·14-s − 61.9·16-s − 92.9·17-s + 13.3·19-s − 66.6·20-s + 69.4·22-s − 219.·23-s − 59.5·25-s + 49.0·28-s + 199.·29-s + 307.·31-s + 257.·32-s + 374.·34-s − 48.1·35-s − 333.·37-s − 53.9·38-s + 7.91·40-s + 200.·41-s + 116.·43-s − 142.·44-s + ⋯
L(s)  = 1  − 1.42·2-s + 1.03·4-s − 0.723·5-s + 0.321·7-s − 0.0432·8-s + 1.03·10-s − 0.472·11-s − 0.457·14-s − 0.968·16-s − 1.32·17-s + 0.161·19-s − 0.745·20-s + 0.673·22-s − 1.99·23-s − 0.476·25-s + 0.331·28-s + 1.27·29-s + 1.78·31-s + 1.42·32-s + 1.88·34-s − 0.232·35-s − 1.48·37-s − 0.230·38-s + 0.0312·40-s + 0.764·41-s + 0.414·43-s − 0.486·44-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: \(0\)
Selberg data: \((2,\ 1521,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.3661060582\)
\(L(\frac12)\) \(\approx\) \(0.3661060582\)
\(L(\frac{5}{2})\) 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 \( 1 + 4.03T + 8T^{2} \)
5 \( 1 + 8.08T + 125T^{2} \)
7 \( 1 - 5.95T + 343T^{2} \)
11 \( 1 + 17.2T + 1.33e3T^{2} \)
17 \( 1 + 92.9T + 4.91e3T^{2} \)
19 \( 1 - 13.3T + 6.85e3T^{2} \)
23 \( 1 + 219.T + 1.21e4T^{2} \)
29 \( 1 - 199.T + 2.43e4T^{2} \)
31 \( 1 - 307.T + 2.97e4T^{2} \)
37 \( 1 + 333.T + 5.06e4T^{2} \)
41 \( 1 - 200.T + 6.89e4T^{2} \)
43 \( 1 - 116.T + 7.95e4T^{2} \)
47 \( 1 + 338.T + 1.03e5T^{2} \)
53 \( 1 - 26.6T + 1.48e5T^{2} \)
59 \( 1 - 280.T + 2.05e5T^{2} \)
61 \( 1 + 207.T + 2.26e5T^{2} \)
67 \( 1 + 285.T + 3.00e5T^{2} \)
71 \( 1 - 317.T + 3.57e5T^{2} \)
73 \( 1 + 63.0T + 3.89e5T^{2} \)
79 \( 1 + 623.T + 4.93e5T^{2} \)
83 \( 1 + 659.T + 5.71e5T^{2} \)
89 \( 1 + 1.27e3T + 7.04e5T^{2} \)
97 \( 1 - 603.T + 9.12e5T^{2} \)
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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

−8.927507097731040900250293326520, −8.219282745228048621731989998786, −7.927969465112682213306124276738, −6.96384254611932788067457240095, −6.17562193915824473386983484990, −4.77733684527698812913825982145, −4.09722504588904597567451141318, −2.64495651826958857405369200278, −1.65907679768762930379043362964, −0.36517527287540022884966710172, 0.36517527287540022884966710172, 1.65907679768762930379043362964, 2.64495651826958857405369200278, 4.09722504588904597567451141318, 4.77733684527698812913825982145, 6.17562193915824473386983484990, 6.96384254611932788067457240095, 7.927969465112682213306124276738, 8.219282745228048621731989998786, 8.927507097731040900250293326520

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