Properties

Label 2-39e2-1.1-c3-0-36
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  − 4.54·2-s + 12.6·4-s − 12.9·5-s + 16.7·7-s − 21.2·8-s + 58.7·10-s − 24.9·11-s − 76.0·14-s − 4.67·16-s + 134.·17-s − 14.9·19-s − 163.·20-s + 113.·22-s − 72·23-s + 41.7·25-s + 212.·28-s + 206.·29-s − 249.·31-s + 191.·32-s − 610.·34-s − 216·35-s + 293.·37-s + 67.9·38-s + 274.·40-s − 250.·41-s + 432.·43-s − 316.·44-s + ⋯
L(s)  = 1  − 1.60·2-s + 1.58·4-s − 1.15·5-s + 0.903·7-s − 0.940·8-s + 1.85·10-s − 0.683·11-s − 1.45·14-s − 0.0731·16-s + 1.91·17-s − 0.180·19-s − 1.83·20-s + 1.09·22-s − 0.652·23-s + 0.333·25-s + 1.43·28-s + 1.32·29-s − 1.44·31-s + 1.05·32-s − 3.07·34-s − 1.04·35-s + 1.30·37-s + 0.289·38-s + 1.08·40-s − 0.954·41-s + 1.53·43-s − 1.08·44-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

\(L(2)\) \(\approx\) \(0.6674431250\)
\(L(\frac12)\) \(\approx\) \(0.6674431250\)
\(L(\frac{5}{2})\) 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 \( 1 + 4.54T + 8T^{2} \)
5 \( 1 + 12.9T + 125T^{2} \)
7 \( 1 - 16.7T + 343T^{2} \)
11 \( 1 + 24.9T + 1.33e3T^{2} \)
17 \( 1 - 134.T + 4.91e3T^{2} \)
19 \( 1 + 14.9T + 6.85e3T^{2} \)
23 \( 1 + 72T + 1.21e4T^{2} \)
29 \( 1 - 206.T + 2.43e4T^{2} \)
31 \( 1 + 249.T + 2.97e4T^{2} \)
37 \( 1 - 293.T + 5.06e4T^{2} \)
41 \( 1 + 250.T + 6.89e4T^{2} \)
43 \( 1 - 432.T + 7.95e4T^{2} \)
47 \( 1 - 159.T + 1.03e5T^{2} \)
53 \( 1 - 194.T + 1.48e5T^{2} \)
59 \( 1 + 232.T + 2.05e5T^{2} \)
61 \( 1 + 185.T + 2.26e5T^{2} \)
67 \( 1 + 39.4T + 3.00e5T^{2} \)
71 \( 1 + 920.T + 3.57e5T^{2} \)
73 \( 1 + 549.T + 3.89e5T^{2} \)
79 \( 1 - 933.T + 4.93e5T^{2} \)
83 \( 1 - 1.09e3T + 5.71e5T^{2} \)
89 \( 1 - 532.T + 7.04e5T^{2} \)
97 \( 1 + 362.T + 9.12e5T^{2} \)
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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

−8.954697438118362192628865652430, −8.123266130715833526068522251775, −7.76847883819536442882545404718, −7.34494006453727997070147313591, −6.03227056226892833532949985796, −4.95161516443529261689380002837, −3.91381682397942018357553417481, −2.71586862987272771262392252407, −1.48814742020961620584734487152, −0.54155455502002189805770486356, 0.54155455502002189805770486356, 1.48814742020961620584734487152, 2.71586862987272771262392252407, 3.91381682397942018357553417481, 4.95161516443529261689380002837, 6.03227056226892833532949985796, 7.34494006453727997070147313591, 7.76847883819536442882545404718, 8.123266130715833526068522251775, 8.954697438118362192628865652430

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