Properties

Label 2-39e2-1.1-c3-0-50
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  + 2.04·2-s − 3.82·4-s + 12.0·5-s − 29.7·7-s − 24.1·8-s + 24.6·10-s + 28.0·11-s − 60.7·14-s − 18.7·16-s + 50.6·17-s − 105.·19-s − 46.2·20-s + 57.3·22-s + 160.·23-s + 20.9·25-s + 113.·28-s − 140.·29-s − 223.·31-s + 155.·32-s + 103.·34-s − 359.·35-s + 228.·37-s − 214.·38-s − 291.·40-s − 295.·41-s + 192.·43-s − 107.·44-s + ⋯
L(s)  = 1  + 0.722·2-s − 0.478·4-s + 1.08·5-s − 1.60·7-s − 1.06·8-s + 0.780·10-s + 0.769·11-s − 1.15·14-s − 0.292·16-s + 0.722·17-s − 1.26·19-s − 0.517·20-s + 0.555·22-s + 1.45·23-s + 0.167·25-s + 0.768·28-s − 0.897·29-s − 1.29·31-s + 0.856·32-s + 0.521·34-s − 1.73·35-s + 1.01·37-s − 0.916·38-s − 1.15·40-s − 1.12·41-s + 0.681·43-s − 0.368·44-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

\(L(2)\) \(\approx\) \(2.257342613\)
\(L(\frac12)\) \(\approx\) \(2.257342613\)
\(L(\frac{5}{2})\) 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 \( 1 - 2.04T + 8T^{2} \)
5 \( 1 - 12.0T + 125T^{2} \)
7 \( 1 + 29.7T + 343T^{2} \)
11 \( 1 - 28.0T + 1.33e3T^{2} \)
17 \( 1 - 50.6T + 4.91e3T^{2} \)
19 \( 1 + 105.T + 6.85e3T^{2} \)
23 \( 1 - 160.T + 1.21e4T^{2} \)
29 \( 1 + 140.T + 2.43e4T^{2} \)
31 \( 1 + 223.T + 2.97e4T^{2} \)
37 \( 1 - 228.T + 5.06e4T^{2} \)
41 \( 1 + 295.T + 6.89e4T^{2} \)
43 \( 1 - 192.T + 7.95e4T^{2} \)
47 \( 1 - 36.9T + 1.03e5T^{2} \)
53 \( 1 + 149.T + 1.48e5T^{2} \)
59 \( 1 - 438.T + 2.05e5T^{2} \)
61 \( 1 - 286.T + 2.26e5T^{2} \)
67 \( 1 - 537.T + 3.00e5T^{2} \)
71 \( 1 - 102.T + 3.57e5T^{2} \)
73 \( 1 - 75.5T + 3.89e5T^{2} \)
79 \( 1 - 17.5T + 4.93e5T^{2} \)
83 \( 1 - 1.46e3T + 5.71e5T^{2} \)
89 \( 1 + 334.T + 7.04e5T^{2} \)
97 \( 1 - 748.T + 9.12e5T^{2} \)
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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

−9.339455300249884140554886718621, −8.622606503510580469599466498147, −7.15813407358746520742914594085, −6.34013696399708797108113476225, −5.88983438179165524559835731941, −5.04827006338090137344452833250, −3.86150361433160298516538873860, −3.28461171069360037778182227195, −2.16452718410463033506869377886, −0.64287455096588829505123079905, 0.64287455096588829505123079905, 2.16452718410463033506869377886, 3.28461171069360037778182227195, 3.86150361433160298516538873860, 5.04827006338090137344452833250, 5.88983438179165524559835731941, 6.34013696399708797108113476225, 7.15813407358746520742914594085, 8.622606503510580469599466498147, 9.339455300249884140554886718621

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