Properties

Label 2-39e2-1.1-c3-0-127
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  + 1.46·2-s − 5.85·4-s − 9.85·5-s + 29.9·7-s − 20.3·8-s − 14.4·10-s − 46.9·11-s + 43.8·14-s + 17.0·16-s + 48.2·17-s + 120.·19-s + 57.6·20-s − 68.7·22-s − 130.·23-s − 27.9·25-s − 175.·28-s + 194.·29-s − 32.0·31-s + 187.·32-s + 70.7·34-s − 294.·35-s − 32.4·37-s + 176.·38-s + 200.·40-s + 241.·41-s + 96.4·43-s + 274.·44-s + ⋯
L(s)  = 1  + 0.518·2-s − 0.731·4-s − 0.881·5-s + 1.61·7-s − 0.897·8-s − 0.456·10-s − 1.28·11-s + 0.837·14-s + 0.266·16-s + 0.688·17-s + 1.45·19-s + 0.644·20-s − 0.666·22-s − 1.18·23-s − 0.223·25-s − 1.18·28-s + 1.24·29-s − 0.185·31-s + 1.03·32-s + 0.356·34-s − 1.42·35-s − 0.144·37-s + 0.752·38-s + 0.790·40-s + 0.921·41-s + 0.341·43-s + 0.940·44-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 \( 1 - 1.46T + 8T^{2} \)
5 \( 1 + 9.85T + 125T^{2} \)
7 \( 1 - 29.9T + 343T^{2} \)
11 \( 1 + 46.9T + 1.33e3T^{2} \)
17 \( 1 - 48.2T + 4.91e3T^{2} \)
19 \( 1 - 120.T + 6.85e3T^{2} \)
23 \( 1 + 130.T + 1.21e4T^{2} \)
29 \( 1 - 194.T + 2.43e4T^{2} \)
31 \( 1 + 32.0T + 2.97e4T^{2} \)
37 \( 1 + 32.4T + 5.06e4T^{2} \)
41 \( 1 - 241.T + 6.89e4T^{2} \)
43 \( 1 - 96.4T + 7.95e4T^{2} \)
47 \( 1 + 539.T + 1.03e5T^{2} \)
53 \( 1 - 152.T + 1.48e5T^{2} \)
59 \( 1 + 327.T + 2.05e5T^{2} \)
61 \( 1 + 98.4T + 2.26e5T^{2} \)
67 \( 1 + 441.T + 3.00e5T^{2} \)
71 \( 1 + 345.T + 3.57e5T^{2} \)
73 \( 1 - 773.T + 3.89e5T^{2} \)
79 \( 1 + 150.T + 4.93e5T^{2} \)
83 \( 1 + 337.T + 5.71e5T^{2} \)
89 \( 1 + 169.T + 7.04e5T^{2} \)
97 \( 1 - 214.T + 9.12e5T^{2} \)
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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

−8.309029116802398372080305134743, −8.021136345663474963516144953087, −7.44347447243537547688743538088, −5.86996071652431033583379712785, −5.15345175896201926381732051056, −4.60797453178107970733814881625, −3.72903590617930109952302730860, −2.71773770013627145978133151951, −1.23411088150481203830956589587, 0, 1.23411088150481203830956589587, 2.71773770013627145978133151951, 3.72903590617930109952302730860, 4.60797453178107970733814881625, 5.15345175896201926381732051056, 5.86996071652431033583379712785, 7.44347447243537547688743538088, 8.021136345663474963516144953087, 8.309029116802398372080305134743

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