Properties

Label 2-39e2-1.1-c3-0-6
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  − 5.36·2-s + 20.7·4-s + 2.69·5-s − 15.2·7-s − 68.5·8-s − 14.4·10-s − 66.8·11-s + 81.5·14-s + 201.·16-s − 4.16·17-s − 26.0·19-s + 56.0·20-s + 358.·22-s − 47.3·23-s − 117.·25-s − 315.·28-s − 257.·29-s − 206.·31-s − 532.·32-s + 22.3·34-s − 40.9·35-s − 175.·37-s + 139.·38-s − 184.·40-s + 156.·41-s + 51.9·43-s − 1.38e3·44-s + ⋯
L(s)  = 1  − 1.89·2-s + 2.59·4-s + 0.241·5-s − 0.820·7-s − 3.03·8-s − 0.457·10-s − 1.83·11-s + 1.55·14-s + 3.14·16-s − 0.0594·17-s − 0.314·19-s + 0.626·20-s + 3.47·22-s − 0.429·23-s − 0.941·25-s − 2.13·28-s − 1.64·29-s − 1.19·31-s − 2.94·32-s + 0.112·34-s − 0.197·35-s − 0.780·37-s + 0.597·38-s − 0.730·40-s + 0.595·41-s + 0.184·43-s − 4.76·44-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

\(L(2)\) \(\approx\) \(0.09282858832\)
\(L(\frac12)\) \(\approx\) \(0.09282858832\)
\(L(\frac{5}{2})\) 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 \( 1 + 5.36T + 8T^{2} \)
5 \( 1 - 2.69T + 125T^{2} \)
7 \( 1 + 15.2T + 343T^{2} \)
11 \( 1 + 66.8T + 1.33e3T^{2} \)
17 \( 1 + 4.16T + 4.91e3T^{2} \)
19 \( 1 + 26.0T + 6.85e3T^{2} \)
23 \( 1 + 47.3T + 1.21e4T^{2} \)
29 \( 1 + 257.T + 2.43e4T^{2} \)
31 \( 1 + 206.T + 2.97e4T^{2} \)
37 \( 1 + 175.T + 5.06e4T^{2} \)
41 \( 1 - 156.T + 6.89e4T^{2} \)
43 \( 1 - 51.9T + 7.95e4T^{2} \)
47 \( 1 + 354.T + 1.03e5T^{2} \)
53 \( 1 - 10.4T + 1.48e5T^{2} \)
59 \( 1 - 445.T + 2.05e5T^{2} \)
61 \( 1 - 119.T + 2.26e5T^{2} \)
67 \( 1 - 22.4T + 3.00e5T^{2} \)
71 \( 1 - 285.T + 3.57e5T^{2} \)
73 \( 1 + 740.T + 3.89e5T^{2} \)
79 \( 1 + 547.T + 4.93e5T^{2} \)
83 \( 1 - 603.T + 5.71e5T^{2} \)
89 \( 1 - 215.T + 7.04e5T^{2} \)
97 \( 1 + 1.44e3T + 9.12e5T^{2} \)
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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

−9.235695637149367062276442396984, −8.325191510381031168749681038569, −7.67835121382867349463344855767, −7.07083835306118543686341353343, −6.07218609074005622978774543587, −5.40148964468771628898345430029, −3.55865268352195665765302709292, −2.52458619512019929331950559126, −1.79271479306014364518367995735, −0.18480224497698183056356409812, 0.18480224497698183056356409812, 1.79271479306014364518367995735, 2.52458619512019929331950559126, 3.55865268352195665765302709292, 5.40148964468771628898345430029, 6.07218609074005622978774543587, 7.07083835306118543686341353343, 7.67835121382867349463344855767, 8.325191510381031168749681038569, 9.235695637149367062276442396984

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