Properties

Label 2-39e2-1.1-c3-0-25
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  − 4.12·2-s + 9·4-s + 3.05·5-s − 6.68·7-s − 4.12·8-s − 12.5·10-s − 32.2·11-s + 27.5·14-s − 55·16-s + 28.8·17-s − 101.·19-s + 27.4·20-s + 132.·22-s + 118.·23-s − 115.·25-s − 60.1·28-s − 160.·29-s − 38.0·31-s + 259.·32-s − 118.·34-s − 20.3·35-s − 327.·37-s + 417.·38-s − 12.5·40-s − 56.0·41-s + 127.·43-s − 290.·44-s + ⋯
L(s)  = 1  − 1.45·2-s + 1.12·4-s + 0.272·5-s − 0.360·7-s − 0.182·8-s − 0.397·10-s − 0.883·11-s + 0.526·14-s − 0.859·16-s + 0.411·17-s − 1.22·19-s + 0.306·20-s + 1.28·22-s + 1.07·23-s − 0.925·25-s − 0.405·28-s − 1.02·29-s − 0.220·31-s + 1.43·32-s − 0.600·34-s − 0.0984·35-s − 1.45·37-s + 1.78·38-s − 0.0497·40-s − 0.213·41-s + 0.453·43-s − 0.994·44-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

\(L(2)\) \(\approx\) \(0.5318408528\)
\(L(\frac12)\) \(\approx\) \(0.5318408528\)
\(L(\frac{5}{2})\) 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 \( 1 + 4.12T + 8T^{2} \)
5 \( 1 - 3.05T + 125T^{2} \)
7 \( 1 + 6.68T + 343T^{2} \)
11 \( 1 + 32.2T + 1.33e3T^{2} \)
17 \( 1 - 28.8T + 4.91e3T^{2} \)
19 \( 1 + 101.T + 6.85e3T^{2} \)
23 \( 1 - 118.T + 1.21e4T^{2} \)
29 \( 1 + 160.T + 2.43e4T^{2} \)
31 \( 1 + 38.0T + 2.97e4T^{2} \)
37 \( 1 + 327.T + 5.06e4T^{2} \)
41 \( 1 + 56.0T + 6.89e4T^{2} \)
43 \( 1 - 127.T + 7.95e4T^{2} \)
47 \( 1 - 517.T + 1.03e5T^{2} \)
53 \( 1 - 695.T + 1.48e5T^{2} \)
59 \( 1 + 656.T + 2.05e5T^{2} \)
61 \( 1 - 701.T + 2.26e5T^{2} \)
67 \( 1 + 57.1T + 3.00e5T^{2} \)
71 \( 1 + 309.T + 3.57e5T^{2} \)
73 \( 1 + 389.T + 3.89e5T^{2} \)
79 \( 1 - 901.T + 4.93e5T^{2} \)
83 \( 1 + 687.T + 5.71e5T^{2} \)
89 \( 1 + 1.07e3T + 7.04e5T^{2} \)
97 \( 1 - 1.75e3T + 9.12e5T^{2} \)
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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

−9.030149940915905097031796152418, −8.518471152298397939358354235427, −7.58300543859358617563331709750, −7.05675908518743857300184197537, −6.01937795329651416388540407111, −5.11461393272137153443034943744, −3.88254955175204368082873557903, −2.59231736481393849662359922981, −1.72010638527191251343467174974, −0.43696791310155239604860135844, 0.43696791310155239604860135844, 1.72010638527191251343467174974, 2.59231736481393849662359922981, 3.88254955175204368082873557903, 5.11461393272137153443034943744, 6.01937795329651416388540407111, 7.05675908518743857300184197537, 7.58300543859358617563331709750, 8.518471152298397939358354235427, 9.030149940915905097031796152418

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