Properties

Label 2-39e2-1.1-c3-0-47
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  + 3.69·2-s + 5.63·4-s − 18.7·5-s + 24.1·7-s − 8.74·8-s − 69.2·10-s − 49.2·11-s + 89.1·14-s − 77.3·16-s + 65.3·17-s − 109.·19-s − 105.·20-s − 181.·22-s + 83.2·23-s + 226.·25-s + 135.·28-s − 4.99·29-s + 255.·31-s − 215.·32-s + 241.·34-s − 452.·35-s − 93.6·37-s − 405.·38-s + 164.·40-s + 67.9·41-s + 142.·43-s − 277.·44-s + ⋯
L(s)  = 1  + 1.30·2-s + 0.703·4-s − 1.67·5-s + 1.30·7-s − 0.386·8-s − 2.19·10-s − 1.35·11-s + 1.70·14-s − 1.20·16-s + 0.931·17-s − 1.32·19-s − 1.18·20-s − 1.76·22-s + 0.755·23-s + 1.81·25-s + 0.917·28-s − 0.0319·29-s + 1.48·31-s − 1.19·32-s + 1.21·34-s − 2.18·35-s − 0.416·37-s − 1.73·38-s + 0.648·40-s + 0.258·41-s + 0.505·43-s − 0.950·44-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

\(L(2)\) \(\approx\) \(2.612429565\)
\(L(\frac12)\) \(\approx\) \(2.612429565\)
\(L(\frac{5}{2})\) 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 \( 1 - 3.69T + 8T^{2} \)
5 \( 1 + 18.7T + 125T^{2} \)
7 \( 1 - 24.1T + 343T^{2} \)
11 \( 1 + 49.2T + 1.33e3T^{2} \)
17 \( 1 - 65.3T + 4.91e3T^{2} \)
19 \( 1 + 109.T + 6.85e3T^{2} \)
23 \( 1 - 83.2T + 1.21e4T^{2} \)
29 \( 1 + 4.99T + 2.43e4T^{2} \)
31 \( 1 - 255.T + 2.97e4T^{2} \)
37 \( 1 + 93.6T + 5.06e4T^{2} \)
41 \( 1 - 67.9T + 6.89e4T^{2} \)
43 \( 1 - 142.T + 7.95e4T^{2} \)
47 \( 1 + 379.T + 1.03e5T^{2} \)
53 \( 1 - 389.T + 1.48e5T^{2} \)
59 \( 1 - 133.T + 2.05e5T^{2} \)
61 \( 1 - 620.T + 2.26e5T^{2} \)
67 \( 1 + 119.T + 3.00e5T^{2} \)
71 \( 1 - 361.T + 3.57e5T^{2} \)
73 \( 1 - 748.T + 3.89e5T^{2} \)
79 \( 1 - 514.T + 4.93e5T^{2} \)
83 \( 1 - 260.T + 5.71e5T^{2} \)
89 \( 1 + 833.T + 7.04e5T^{2} \)
97 \( 1 - 1.48e3T + 9.12e5T^{2} \)
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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

−8.668602270343548237966909370655, −8.162018182743891762388026341111, −7.57118439235306853154415042846, −6.59949636248297481535327904345, −5.35987706852774186637700628969, −4.82869935313299978349147121840, −4.19059980242903861843967853304, −3.32300286958783656611670449928, −2.36837783298885919606416675926, −0.63033876686140353323652452227, 0.63033876686140353323652452227, 2.36837783298885919606416675926, 3.32300286958783656611670449928, 4.19059980242903861843967853304, 4.82869935313299978349147121840, 5.35987706852774186637700628969, 6.59949636248297481535327904345, 7.57118439235306853154415042846, 8.162018182743891762388026341111, 8.668602270343548237966909370655

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