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Properties

Label 2-1287-1287.835-c0-0-0
Degree $2$
Conductor $1287$
Sign $0.815 - 0.578i$
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  − 3-s + 4-s + (0.5 + 0.866i)5-s + (−0.866 + 0.5i)7-s + 9-s − i·11-s − 12-s − i·13-s + (−0.5 − 0.866i)15-s + 16-s + (0.866 + 0.5i)17-s + (0.866 + 0.5i)19-s + (0.5 + 0.866i)20-s + (0.866 − 0.5i)21-s + (−0.5 + 0.866i)23-s + ⋯
L(s)  = 1  − 3-s + 4-s + (0.5 + 0.866i)5-s + (−0.866 + 0.5i)7-s + 9-s − i·11-s − 12-s − i·13-s + (−0.5 − 0.866i)15-s + 16-s + (0.866 + 0.5i)17-s + (0.866 + 0.5i)19-s + (0.5 + 0.866i)20-s + (0.866 − 0.5i)21-s + (−0.5 + 0.866i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.815 - 0.578i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.815 - 0.578i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $0.815 - 0.578i$
Analytic conductor: \(0.642296\)
Root analytic conductor: \(0.801434\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1287} (835, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1287,\ (\ :0),\ 0.815 - 0.578i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.034195154\)
\(L(\frac12)\) \(\approx\) \(1.034195154\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
11 \( 1 + iT \)
13 \( 1 + iT \)
good2 \( 1 - T^{2} \)
5 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
7 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
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 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 - 2iT - T^{2} \)
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 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
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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

−10.16547546015946568172134300642, −9.502862206723730346259623424084, −8.053278901364916001893834846944, −7.27685821389742211977783119372, −6.43973335062187485555968963896, −5.80241883174070854400221438260, −5.49412772802313831078773627695, −3.44784570856421545828200925238, −2.99667949594355469019223986233, −1.47336242289764262725880532539, 1.15491495898429583127451021253, 2.31833016133326040260263770400, 3.83982966316452379101846228204, 4.84543874811346221593563499074, 5.66329180018769904879876546937, 6.56561831994785383894942303113, 7.03802035479903233998184716497, 7.88857033181670214772062786552, 9.465261516613688786863038101329, 9.728040237598300051135781239912

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