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Properties

Label 2-1287-1287.1264-c0-0-1
Degree $2$
Conductor $1287$
Sign $-0.568 + 0.822i$
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  + (0.5 − 0.866i)3-s + (−0.5 − 0.866i)4-s + (0.5 − 0.866i)5-s − i·7-s + (−0.499 − 0.866i)9-s + (0.866 + 0.5i)11-s − 0.999·12-s + (−0.866 + 0.5i)13-s + (−0.499 − 0.866i)15-s + (−0.499 + 0.866i)16-s + (0.866 + 0.5i)17-s + (0.866 + 0.5i)19-s − 0.999·20-s + (−0.866 − 0.5i)21-s + 23-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)3-s + (−0.5 − 0.866i)4-s + (0.5 − 0.866i)5-s − i·7-s + (−0.499 − 0.866i)9-s + (0.866 + 0.5i)11-s − 0.999·12-s + (−0.866 + 0.5i)13-s + (−0.499 − 0.866i)15-s + (−0.499 + 0.866i)16-s + (0.866 + 0.5i)17-s + (0.866 + 0.5i)19-s − 0.999·20-s + (−0.866 − 0.5i)21-s + 23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.568 + 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.568 + 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $-0.568 + 0.822i$
Analytic conductor: \(0.642296\)
Root analytic conductor: \(0.801434\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1287} (1264, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1287,\ (\ :0),\ -0.568 + 0.822i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.232144586\)
\(L(\frac12)\) \(\approx\) \(1.232144586\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + (-0.866 - 0.5i)T \)
13 \( 1 + (0.866 - 0.5i)T \)
good2 \( 1 + (0.5 + 0.866i)T^{2} \)
5 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
7 \( 1 + iT - T^{2} \)
17 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
23 \( 1 - T + T^{2} \)
29 \( 1 + (1.73 + i)T + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 - iT - T^{2} \)
43 \( 1 - iT - T^{2} \)
47 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + iT - T^{2} \)
67 \( 1 + T + T^{2} \)
71 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
89 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 - T + T^{2} \)
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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

−9.562991585285932689141769927222, −8.977911122570370094087615637513, −7.896217400761290188464828547643, −7.16589398490219261997396560826, −6.31088634539373859357618005381, −5.37990334083587375773434503036, −4.52355382343400520020966575399, −3.47872133900900992813000905752, −1.71326617048426051152931158465, −1.16905796660583495902150279351, 2.47020962831691831250803947116, 3.07539682180112874128978521843, 3.86638498920024900584717128510, 5.18015505353844336756382200541, 5.65979299599927104035211279314, 7.14868866823476633364515473715, 7.68292453288654847516603239886, 8.951933743984817935298252684505, 9.144579819720679780490670821840, 9.865944661642601476919361237520

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