Properties

Label 2-39e2-1.1-c3-0-166
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  + 4.20·2-s + 9.71·4-s − 11.4·5-s + 11.2·7-s + 7.22·8-s − 48.1·10-s + 25.8·11-s + 47.3·14-s − 47.3·16-s + 20.3·17-s − 154.·19-s − 111.·20-s + 108.·22-s + 180.·23-s + 5.69·25-s + 109.·28-s + 20.4·29-s − 266.·31-s − 256.·32-s + 85.5·34-s − 128.·35-s − 115.·37-s − 651.·38-s − 82.5·40-s + 391.·41-s + 151.·43-s + 251.·44-s + ⋯
L(s)  = 1  + 1.48·2-s + 1.21·4-s − 1.02·5-s + 0.607·7-s + 0.319·8-s − 1.52·10-s + 0.709·11-s + 0.904·14-s − 0.739·16-s + 0.290·17-s − 1.86·19-s − 1.24·20-s + 1.05·22-s + 1.63·23-s + 0.0455·25-s + 0.738·28-s + 0.130·29-s − 1.54·31-s − 1.41·32-s + 0.431·34-s − 0.621·35-s − 0.515·37-s − 2.77·38-s − 0.326·40-s + 1.49·41-s + 0.536·43-s + 0.861·44-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 \( 1 - 4.20T + 8T^{2} \)
5 \( 1 + 11.4T + 125T^{2} \)
7 \( 1 - 11.2T + 343T^{2} \)
11 \( 1 - 25.8T + 1.33e3T^{2} \)
17 \( 1 - 20.3T + 4.91e3T^{2} \)
19 \( 1 + 154.T + 6.85e3T^{2} \)
23 \( 1 - 180.T + 1.21e4T^{2} \)
29 \( 1 - 20.4T + 2.43e4T^{2} \)
31 \( 1 + 266.T + 2.97e4T^{2} \)
37 \( 1 + 115.T + 5.06e4T^{2} \)
41 \( 1 - 391.T + 6.89e4T^{2} \)
43 \( 1 - 151.T + 7.95e4T^{2} \)
47 \( 1 + 467.T + 1.03e5T^{2} \)
53 \( 1 + 79.9T + 1.48e5T^{2} \)
59 \( 1 + 873.T + 2.05e5T^{2} \)
61 \( 1 + 187.T + 2.26e5T^{2} \)
67 \( 1 - 609.T + 3.00e5T^{2} \)
71 \( 1 - 248.T + 3.57e5T^{2} \)
73 \( 1 + 852.T + 3.89e5T^{2} \)
79 \( 1 + 331.T + 4.93e5T^{2} \)
83 \( 1 + 435.T + 5.71e5T^{2} \)
89 \( 1 - 259.T + 7.04e5T^{2} \)
97 \( 1 + 1.22e3T + 9.12e5T^{2} \)
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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

−8.629115461214584515797831362494, −7.70017653773330940899221171091, −6.88573615103041936550465959978, −6.12633880183201989676191793691, −5.11821223993511495805119607344, −4.38352851836471252705677979289, −3.82613937378273430647774141373, −2.89990960219841569220249167622, −1.64012638054825888644664733445, 0, 1.64012638054825888644664733445, 2.89990960219841569220249167622, 3.82613937378273430647774141373, 4.38352851836471252705677979289, 5.11821223993511495805119607344, 6.12633880183201989676191793691, 6.88573615103041936550465959978, 7.70017653773330940899221171091, 8.629115461214584515797831362494

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