Properties

Label 2-39e2-1.1-c3-0-126
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  − 1.56·2-s − 5.56·4-s − 3.56·5-s + 27.1·7-s + 21.1·8-s + 5.56·10-s + 15.2·11-s − 42.4·14-s + 11.4·16-s − 44.5·17-s − 23.9·19-s + 19.8·20-s − 23.8·22-s − 122.·23-s − 112.·25-s − 151.·28-s + 219.·29-s − 27.0·31-s − 187.·32-s + 69.5·34-s − 96.7·35-s − 94.1·37-s + 37.4·38-s − 75.4·40-s − 160.·41-s − 151.·43-s − 84.8·44-s + ⋯
L(s)  = 1  − 0.552·2-s − 0.695·4-s − 0.318·5-s + 1.46·7-s + 0.935·8-s + 0.175·10-s + 0.418·11-s − 0.810·14-s + 0.178·16-s − 0.635·17-s − 0.289·19-s + 0.221·20-s − 0.230·22-s − 1.11·23-s − 0.898·25-s − 1.02·28-s + 1.40·29-s − 0.156·31-s − 1.03·32-s + 0.350·34-s − 0.467·35-s − 0.418·37-s + 0.159·38-s − 0.298·40-s − 0.610·41-s − 0.536·43-s − 0.290·44-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 \( 1 + 1.56T + 8T^{2} \)
5 \( 1 + 3.56T + 125T^{2} \)
7 \( 1 - 27.1T + 343T^{2} \)
11 \( 1 - 15.2T + 1.33e3T^{2} \)
17 \( 1 + 44.5T + 4.91e3T^{2} \)
19 \( 1 + 23.9T + 6.85e3T^{2} \)
23 \( 1 + 122.T + 1.21e4T^{2} \)
29 \( 1 - 219.T + 2.43e4T^{2} \)
31 \( 1 + 27.0T + 2.97e4T^{2} \)
37 \( 1 + 94.1T + 5.06e4T^{2} \)
41 \( 1 + 160.T + 6.89e4T^{2} \)
43 \( 1 + 151.T + 7.95e4T^{2} \)
47 \( 1 - 466.T + 1.03e5T^{2} \)
53 \( 1 - 120.T + 1.48e5T^{2} \)
59 \( 1 + 439.T + 2.05e5T^{2} \)
61 \( 1 + 137.T + 2.26e5T^{2} \)
67 \( 1 + 512.T + 3.00e5T^{2} \)
71 \( 1 - 410.T + 3.57e5T^{2} \)
73 \( 1 - 308.T + 3.89e5T^{2} \)
79 \( 1 + 586.T + 4.93e5T^{2} \)
83 \( 1 - 1.35e3T + 5.71e5T^{2} \)
89 \( 1 - 439.T + 7.04e5T^{2} \)
97 \( 1 - 1.51e3T + 9.12e5T^{2} \)
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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

−8.598454228163027180398436626523, −8.078916271464225045122566790217, −7.44753963865920421021343975895, −6.30344870591967083347920950880, −5.17491368247831587147308559098, −4.47698345894573715125797139923, −3.81433886243805089861277327875, −2.13980044257665512243259396753, −1.23026557381232837221757028052, 0, 1.23026557381232837221757028052, 2.13980044257665512243259396753, 3.81433886243805089861277327875, 4.47698345894573715125797139923, 5.17491368247831587147308559098, 6.30344870591967083347920950880, 7.44753963865920421021343975895, 8.078916271464225045122566790217, 8.598454228163027180398436626523

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