Properties

Label 2-12882-1.1-c1-0-1
Degree $2$
Conductor $12882$
Sign $1$
Analytic cond. $102.863$
Root an. cond. $10.1421$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 4-s − 3·5-s + 6-s + 3·7-s + 8-s + 9-s − 3·10-s + 12-s − 2·13-s + 3·14-s − 3·15-s + 16-s − 4·17-s + 18-s − 19-s − 3·20-s + 3·21-s + 2·23-s + 24-s + 4·25-s − 2·26-s + 27-s + 3·28-s − 6·29-s − 3·30-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.34·5-s + 0.408·6-s + 1.13·7-s + 0.353·8-s + 1/3·9-s − 0.948·10-s + 0.288·12-s − 0.554·13-s + 0.801·14-s − 0.774·15-s + 1/4·16-s − 0.970·17-s + 0.235·18-s − 0.229·19-s − 0.670·20-s + 0.654·21-s + 0.417·23-s + 0.204·24-s + 4/5·25-s − 0.392·26-s + 0.192·27-s + 0.566·28-s − 1.11·29-s − 0.547·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12882\)    =    \(2 \cdot 3 \cdot 19 \cdot 113\)
Sign: $1$
Analytic conductor: \(102.863\)
Root analytic conductor: \(10.1421\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 12882,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.571038663\)
\(L(\frac12)\) \(\approx\) \(3.571038663\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 - T \)
19 \( 1 + T \)
113 \( 1 - T \)
good5 \( 1 + 3 T + p T^{2} \)
7 \( 1 - 3 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 12 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 9 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−16.05189050176568, −15.43632221020996, −15.12979598225287, −14.59793271735065, −14.26213315972199, −13.36056442759319, −12.99671808240849, −12.27679275771018, −11.68763422553832, −11.23437272495146, −10.90273001183021, −9.966143333186743, −9.210577003334708, −8.478901908424891, −7.921993405287132, −7.607640784448587, −6.906560618059118, −6.174263180874100, −5.151815933618790, −4.524948745911543, −4.237211141219863, −3.430648930350976, −2.599644777825019, −1.917544733012250, −0.7328670360679781, 0.7328670360679781, 1.917544733012250, 2.599644777825019, 3.430648930350976, 4.237211141219863, 4.524948745911543, 5.151815933618790, 6.174263180874100, 6.906560618059118, 7.607640784448587, 7.921993405287132, 8.478901908424891, 9.210577003334708, 9.966143333186743, 10.90273001183021, 11.23437272495146, 11.68763422553832, 12.27679275771018, 12.99671808240849, 13.36056442759319, 14.26213315972199, 14.59793271735065, 15.12979598225287, 15.43632221020996, 16.05189050176568

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