Properties

Label 2-39e2-1.1-c3-0-94
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  + 1.22·2-s − 6.48·4-s − 17.3·5-s − 19.1·7-s − 17.8·8-s − 21.2·10-s + 32.6·11-s − 23.5·14-s + 30.0·16-s + 86.5·17-s + 76.0·19-s + 112.·20-s + 40.1·22-s − 75.3·23-s + 174.·25-s + 124.·28-s − 286.·29-s + 200.·31-s + 179.·32-s + 106.·34-s + 332.·35-s − 173.·37-s + 93.4·38-s + 308.·40-s + 496.·41-s − 187.·43-s − 211.·44-s + ⋯
L(s)  = 1  + 0.434·2-s − 0.811·4-s − 1.54·5-s − 1.03·7-s − 0.786·8-s − 0.672·10-s + 0.895·11-s − 0.450·14-s + 0.469·16-s + 1.23·17-s + 0.918·19-s + 1.25·20-s + 0.388·22-s − 0.682·23-s + 1.39·25-s + 0.840·28-s − 1.83·29-s + 1.16·31-s + 0.990·32-s + 0.536·34-s + 1.60·35-s − 0.769·37-s + 0.399·38-s + 1.21·40-s + 1.88·41-s − 0.664·43-s − 0.726·44-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 \( 1 - 1.22T + 8T^{2} \)
5 \( 1 + 17.3T + 125T^{2} \)
7 \( 1 + 19.1T + 343T^{2} \)
11 \( 1 - 32.6T + 1.33e3T^{2} \)
17 \( 1 - 86.5T + 4.91e3T^{2} \)
19 \( 1 - 76.0T + 6.85e3T^{2} \)
23 \( 1 + 75.3T + 1.21e4T^{2} \)
29 \( 1 + 286.T + 2.43e4T^{2} \)
31 \( 1 - 200.T + 2.97e4T^{2} \)
37 \( 1 + 173.T + 5.06e4T^{2} \)
41 \( 1 - 496.T + 6.89e4T^{2} \)
43 \( 1 + 187.T + 7.95e4T^{2} \)
47 \( 1 - 254.T + 1.03e5T^{2} \)
53 \( 1 + 63.6T + 1.48e5T^{2} \)
59 \( 1 + 67.9T + 2.05e5T^{2} \)
61 \( 1 - 400.T + 2.26e5T^{2} \)
67 \( 1 - 510.T + 3.00e5T^{2} \)
71 \( 1 - 200.T + 3.57e5T^{2} \)
73 \( 1 - 168.T + 3.89e5T^{2} \)
79 \( 1 + 1.11e3T + 4.93e5T^{2} \)
83 \( 1 + 724.T + 5.71e5T^{2} \)
89 \( 1 + 821.T + 7.04e5T^{2} \)
97 \( 1 - 19.5T + 9.12e5T^{2} \)
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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

−8.696176113986554354419965763499, −7.85684144901107565282400366916, −7.20327546420206312753688182546, −6.13480133415078751153534441828, −5.32544964686202716460846126836, −4.14656200567687772427498275018, −3.72305010059105517057641520439, −3.04807893999205490025972870354, −0.962251394750458867003509367429, 0, 0.962251394750458867003509367429, 3.04807893999205490025972870354, 3.72305010059105517057641520439, 4.14656200567687772427498275018, 5.32544964686202716460846126836, 6.13480133415078751153534441828, 7.20327546420206312753688182546, 7.85684144901107565282400366916, 8.696176113986554354419965763499

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