Properties

Label 2-1287-143.142-c0-0-3
Degree $2$
Conductor $1287$
Sign $0.707 - 0.707i$
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  + 1.41·2-s + 1.00·4-s + 1.41i·5-s + 2.00i·10-s + (0.707 + 0.707i)11-s i·13-s − 0.999·16-s + 1.41i·20-s + (1.00 + 1.00i)22-s − 1.00·25-s − 1.41i·26-s − 1.41·32-s + 1.41·41-s + (0.707 + 0.707i)44-s − 1.41i·47-s + ⋯
L(s)  = 1  + 1.41·2-s + 1.00·4-s + 1.41i·5-s + 2.00i·10-s + (0.707 + 0.707i)11-s i·13-s − 0.999·16-s + 1.41i·20-s + (1.00 + 1.00i)22-s − 1.00·25-s − 1.41i·26-s − 1.41·32-s + 1.41·41-s + (0.707 + 0.707i)44-s − 1.41i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $0.707 - 0.707i$
Analytic conductor: \(0.642296\)
Root analytic conductor: \(0.801434\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1287} (1000, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1287,\ (\ :0),\ 0.707 - 0.707i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.120981772\)
\(L(\frac12)\) \(\approx\) \(2.120981772\)
\(L(1)\) 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 \( 1 + (-0.707 - 0.707i)T \)
13 \( 1 + iT \)
good2 \( 1 - 1.41T + T^{2} \)
5 \( 1 - 1.41iT - T^{2} \)
7 \( 1 + T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - 1.41T + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + 1.41iT - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + 1.41iT - T^{2} \)
61 \( 1 + 2iT - T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - 1.41iT - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + 2iT - T^{2} \)
83 \( 1 + 1.41T + T^{2} \)
89 \( 1 - 1.41iT - T^{2} \)
97 \( 1 - T^{2} \)
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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

−10.13147666529101473514834987981, −9.353890148339358385011275156057, −8.085145865242392571968158165167, −7.10089717010143671546310084791, −6.55071405959281901704079285525, −5.76913589984929420473186488260, −4.82657331814778032184019446496, −3.80099604081656560448266874899, −3.13782929097991862396814459764, −2.19849562574212722740814890715, 1.39101788316382092332567372647, 2.85586617580369934813398021190, 4.15145549029434187090176070890, 4.40550391454081969945127358542, 5.47653976344901520617722588143, 6.07281777113144528444814171380, 7.00790473567206659989957567947, 8.230157115431977138748214605092, 9.057539118186760601728593504993, 9.438063365195954507601409502679

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