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Properties

Label 2-287-287.286-c2-0-16
Degree $2$
Conductor $287$
Sign $-0.0919 - 0.995i$
Analytic cond. $7.82018$
Root an. cond. $2.79645$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.89·2-s + 0.529·3-s − 0.393·4-s + 4.90i·5-s + 1.00·6-s + (5.71 + 4.03i)7-s − 8.34·8-s − 8.71·9-s + 9.30i·10-s + 15.0i·11-s − 0.208·12-s − 4.92·13-s + (10.8 + 7.66i)14-s + 2.59i·15-s − 14.2·16-s + 10.9·17-s + ⋯
L(s)  = 1  + 0.949·2-s + 0.176·3-s − 0.0983·4-s + 0.980i·5-s + 0.167·6-s + (0.817 + 0.576i)7-s − 1.04·8-s − 0.968·9-s + 0.930i·10-s + 1.36i·11-s − 0.0173·12-s − 0.378·13-s + (0.775 + 0.547i)14-s + 0.172i·15-s − 0.891·16-s + 0.646·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0919 - 0.995i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0919 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $-0.0919 - 0.995i$
Analytic conductor: \(7.82018\)
Root analytic conductor: \(2.79645\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (286, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 287,\ (\ :1),\ -0.0919 - 0.995i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.42892 + 1.56696i\)
\(L(\frac12)\) \(\approx\) \(1.42892 + 1.56696i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-5.71 - 4.03i)T \)
41 \( 1 + (-26.6 - 31.1i)T \)
good2 \( 1 - 1.89T + 4T^{2} \)
3 \( 1 - 0.529T + 9T^{2} \)
5 \( 1 - 4.90iT - 25T^{2} \)
11 \( 1 - 15.0iT - 121T^{2} \)
13 \( 1 + 4.92T + 169T^{2} \)
17 \( 1 - 10.9T + 289T^{2} \)
19 \( 1 - 14.1T + 361T^{2} \)
23 \( 1 - 6.57T + 529T^{2} \)
29 \( 1 + 20.5iT - 841T^{2} \)
31 \( 1 - 31.6iT - 961T^{2} \)
37 \( 1 + 48.4T + 1.36e3T^{2} \)
43 \( 1 - 16.8T + 1.84e3T^{2} \)
47 \( 1 - 56.3T + 2.20e3T^{2} \)
53 \( 1 + 72.9iT - 2.80e3T^{2} \)
59 \( 1 + 114. iT - 3.48e3T^{2} \)
61 \( 1 + 39.7iT - 3.72e3T^{2} \)
67 \( 1 - 73.8iT - 4.48e3T^{2} \)
71 \( 1 - 25.9iT - 5.04e3T^{2} \)
73 \( 1 + 96.0iT - 5.32e3T^{2} \)
79 \( 1 - 21.5iT - 6.24e3T^{2} \)
83 \( 1 - 80.0iT - 6.88e3T^{2} \)
89 \( 1 + 62.7T + 7.92e3T^{2} \)
97 \( 1 + 58.3T + 9.40e3T^{2} \)
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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

−11.98859625476168259798072017138, −11.20975052277534797229235737313, −9.984428858304412874099455571884, −9.016430600144996886170204315899, −7.898058821319436871992992295246, −6.79280950443660991516365130197, −5.54738771846112020194482408342, −4.81384993867089048854139011585, −3.37058144331521940498316047406, −2.36595050780828619249332588812, 0.78364977466996551664779285053, 3.00663984401518822741962306116, 4.13992987390960583752364857937, 5.28144825407427392163471713593, 5.76174510024580132732553576239, 7.54915481954445164224028881454, 8.643046255290084269037179322269, 9.083305633993174347321355879009, 10.65279164215968147656571795570, 11.65390229107202867307127266235

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