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Properties

Label 2-1287-143.90-c0-0-0
Degree $2$
Conductor $1287$
Sign $-0.434 - 0.900i$
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  + (−0.280 + 0.863i)2-s + (0.142 + 0.103i)4-s + (−0.297 + 0.0966i)5-s + (−0.863 + 0.627i)8-s − 0.284i·10-s + (0.891 + 0.453i)11-s + (0.951 + 0.309i)13-s + (−0.245 − 0.754i)16-s + (−0.0522 − 0.0169i)20-s + (−0.642 + 0.642i)22-s + (−0.729 + 0.530i)25-s + (−0.533 + 0.734i)26-s − 0.346·32-s + (0.196 − 0.270i)40-s + (−1.14 + 0.831i)41-s + ⋯
L(s)  = 1  + (−0.280 + 0.863i)2-s + (0.142 + 0.103i)4-s + (−0.297 + 0.0966i)5-s + (−0.863 + 0.627i)8-s − 0.284i·10-s + (0.891 + 0.453i)11-s + (0.951 + 0.309i)13-s + (−0.245 − 0.754i)16-s + (−0.0522 − 0.0169i)20-s + (−0.642 + 0.642i)22-s + (−0.729 + 0.530i)25-s + (−0.533 + 0.734i)26-s − 0.346·32-s + (0.196 − 0.270i)40-s + (−1.14 + 0.831i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.434 - 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.434 - 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $-0.434 - 0.900i$
Analytic conductor: \(0.642296\)
Root analytic conductor: \(0.801434\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1287} (1234, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1287,\ (\ :0),\ -0.434 - 0.900i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9754551906\)
\(L(\frac12)\) \(\approx\) \(0.9754551906\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 + (-0.891 - 0.453i)T \)
13 \( 1 + (-0.951 - 0.309i)T \)
good2 \( 1 + (0.280 - 0.863i)T + (-0.809 - 0.587i)T^{2} \)
5 \( 1 + (0.297 - 0.0966i)T + (0.809 - 0.587i)T^{2} \)
7 \( 1 + (0.309 + 0.951i)T^{2} \)
17 \( 1 + (0.809 - 0.587i)T^{2} \)
19 \( 1 + (0.309 - 0.951i)T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + (-0.309 - 0.951i)T^{2} \)
31 \( 1 + (0.809 + 0.587i)T^{2} \)
37 \( 1 + (-0.309 - 0.951i)T^{2} \)
41 \( 1 + (1.14 - 0.831i)T + (0.309 - 0.951i)T^{2} \)
43 \( 1 - 1.17iT - T^{2} \)
47 \( 1 + (-0.533 - 0.734i)T + (-0.309 + 0.951i)T^{2} \)
53 \( 1 + (-0.809 - 0.587i)T^{2} \)
59 \( 1 + (-1.04 + 1.44i)T + (-0.309 - 0.951i)T^{2} \)
61 \( 1 + (0.587 - 0.190i)T + (0.809 - 0.587i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (-1.34 + 0.437i)T + (0.809 - 0.587i)T^{2} \)
73 \( 1 + (0.309 + 0.951i)T^{2} \)
79 \( 1 + (-0.587 - 0.190i)T + (0.809 + 0.587i)T^{2} \)
83 \( 1 + (-0.0966 - 0.297i)T + (-0.809 + 0.587i)T^{2} \)
89 \( 1 + 1.41iT - T^{2} \)
97 \( 1 + (0.809 + 0.587i)T^{2} \)
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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

−9.840732330163134674220509886588, −9.139268082763574149576686789193, −8.330945468650909492402339283160, −7.70288308207463406788603407557, −6.73985425704606144174956149033, −6.34228156039065224718763724929, −5.29331349632604396446919642455, −4.09094784393989946124133240960, −3.19049690328518243197324266228, −1.74106556816453233049171766361, 0.987535539520318366802433135556, 2.17332434967664430056988954627, 3.43111050166300276988529382436, 4.01133532205184470111770083322, 5.50923697016820290558272063152, 6.28401206697511331679176102280, 7.07238466390590728951345443452, 8.280883142212281308061541813647, 8.897598443493836823113532171549, 9.698626953263280383204222564121

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