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Properties

Label 2-1287-143.90-c0-0-1
Degree $2$
Conductor $1287$
Sign $0.900 - 0.434i$
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  + (−0.550 + 1.69i)2-s + (−1.76 − 1.27i)4-s + (−1.87 + 0.610i)5-s + (1.69 − 1.23i)8-s − 3.52i·10-s + (0.453 − 0.891i)11-s + (−0.951 − 0.309i)13-s + (0.481 + 1.48i)16-s + (4.08 + 1.32i)20-s + (1.26 + 1.26i)22-s + (2.34 − 1.70i)25-s + (1.04 − 1.44i)26-s − 0.680·32-s + (−2.43 + 3.34i)40-s + (1.14 − 0.831i)41-s + ⋯
L(s)  = 1  + (−0.550 + 1.69i)2-s + (−1.76 − 1.27i)4-s + (−1.87 + 0.610i)5-s + (1.69 − 1.23i)8-s − 3.52i·10-s + (0.453 − 0.891i)11-s + (−0.951 − 0.309i)13-s + (0.481 + 1.48i)16-s + (4.08 + 1.32i)20-s + (1.26 + 1.26i)22-s + (2.34 − 1.70i)25-s + (1.04 − 1.44i)26-s − 0.680·32-s + (−2.43 + 3.34i)40-s + (1.14 − 0.831i)41-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $0.900 - 0.434i$
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: \((2,\ 1287,\ (\ :0),\ 0.900 - 0.434i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3140422896\)
\(L(\frac12)\) \(\approx\) \(0.3140422896\)
\(L(1)\) 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 \( 1 + (-0.453 + 0.891i)T \)
13 \( 1 + (0.951 + 0.309i)T \)
good2 \( 1 + (0.550 - 1.69i)T + (-0.809 - 0.587i)T^{2} \)
5 \( 1 + (1.87 - 0.610i)T + (0.809 - 0.587i)T^{2} \)
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 \( 1 + (1.04 + 1.44i)T + (-0.309 + 0.951i)T^{2} \)
53 \( 1 + (-0.809 - 0.587i)T^{2} \)
59 \( 1 + (0.533 - 0.734i)T + (-0.309 - 0.951i)T^{2} \)
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 \( 1 + (0.610 + 1.87i)T + (-0.809 + 0.587i)T^{2} \)
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

−9.628363367025599779391934453473, −8.607920727289982564691110273673, −8.178280828560914127495964373157, −7.41057656097306994581298809089, −6.95065936055571413659464942216, −6.07234982459476614602337155222, −4.98976870259895955252213600962, −4.13120389870120821409560839311, −3.11788218335191561962320142294, −0.38792450324627204925367081143, 1.17291751408793397789065446316, 2.55635163728696700221799497631, 3.64303918994019518812296849150, 4.33376052105568533463759882314, 4.90533436072248347638917020889, 6.91530672586101765137272511560, 7.77304513140671151124043293232, 8.317168298305545780515846934089, 9.300495252310200357801952257211, 9.682863007538585025470546341215

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