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Properties

Label 2-1287-1287.571-c0-0-5
Degree $2$
Conductor $1287$
Sign $-0.882 + 0.469i$
Analytic cond. $0.642296$
Root an. cond. $0.801434$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.913 − 1.58i)2-s + (−0.104 + 0.994i)3-s + (−1.16 + 2.02i)4-s + (1.66 − 0.743i)6-s + (−0.309 − 0.535i)7-s + 2.44·8-s + (−0.978 − 0.207i)9-s + (−0.5 − 0.866i)11-s + (−1.89 − 1.37i)12-s + (−0.5 + 0.866i)13-s + (−0.564 + 0.977i)14-s + (−1.06 − 1.84i)16-s + (0.564 + 1.73i)18-s − 0.209·19-s + (0.564 − 0.251i)21-s + (−0.913 + 1.58i)22-s + ⋯
L(s)  = 1  + (−0.913 − 1.58i)2-s + (−0.104 + 0.994i)3-s + (−1.16 + 2.02i)4-s + (1.66 − 0.743i)6-s + (−0.309 − 0.535i)7-s + 2.44·8-s + (−0.978 − 0.207i)9-s + (−0.5 − 0.866i)11-s + (−1.89 − 1.37i)12-s + (−0.5 + 0.866i)13-s + (−0.564 + 0.977i)14-s + (−1.06 − 1.84i)16-s + (0.564 + 1.73i)18-s − 0.209·19-s + (0.564 − 0.251i)21-s + (−0.913 + 1.58i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.882 + 0.469i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.882 + 0.469i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $-0.882 + 0.469i$
Analytic conductor: \(0.642296\)
Root analytic conductor: \(0.801434\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1287} (571, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1287,\ (\ :0),\ -0.882 + 0.469i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3606743922\)
\(L(\frac12)\) \(\approx\) \(0.3606743922\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.104 - 0.994i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (0.913 + 1.58i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + (0.5 + 0.866i)T^{2} \)
7 \( 1 + (0.309 + 0.535i)T + (-0.5 + 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + 0.209T + T^{2} \)
23 \( 1 + (-0.978 + 1.69i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (-0.978 + 1.69i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + 1.61T + T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + 1.61T + T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.669 + 1.15i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.5 - 0.866i)T^{2} \)
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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

−9.769415341491069400500640024705, −8.868217205829388385953148650689, −8.545491504352631629957818375132, −7.40372937049465144492788046984, −6.15287412894538294186262632476, −4.74102250998347851002256365794, −4.07912111432806171060840395640, −3.13652282183661720947565087694, −2.31467286507641991163158370721, −0.43806614243204788822796065741, 1.42954246995758992643919477668, 2.90113926422931181796922457117, 4.90141098090848700790354485625, 5.57906053503937587939028313838, 6.22439486644203287807505601455, 7.24437764922002617054802669256, 7.57247890929843910505648501693, 8.253296406738041754953918008689, 9.285957465976034902275246703115, 9.670570991320452710587567476173

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