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Properties

Label 2-287-287.286-c2-0-5
Degree $2$
Conductor $287$
Sign $-0.844 - 0.534i$
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  − 2.67·2-s + 2.89·3-s + 3.15·4-s + 3.34i·5-s − 7.74·6-s + (−4.11 − 5.65i)7-s + 2.26·8-s − 0.605·9-s − 8.94i·10-s + 15.3i·11-s + 9.13·12-s − 7.80·13-s + (11.0 + 15.1i)14-s + 9.68i·15-s − 18.6·16-s − 8.48·17-s + ⋯
L(s)  = 1  − 1.33·2-s + 0.965·3-s + 0.787·4-s + 0.668i·5-s − 1.29·6-s + (−0.588 − 0.808i)7-s + 0.283·8-s − 0.0672·9-s − 0.894i·10-s + 1.39i·11-s + 0.761·12-s − 0.600·13-s + (0.786 + 1.08i)14-s + 0.645i·15-s − 1.16·16-s − 0.499·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.844 - 0.534i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.844 - 0.534i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $-0.844 - 0.534i$
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: \((2,\ 287,\ (\ :1),\ -0.844 - 0.534i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.106994 + 0.369070i\)
\(L(\frac12)\) \(\approx\) \(0.106994 + 0.369070i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (4.11 + 5.65i)T \)
41 \( 1 + (38.1 - 15.1i)T \)
good2 \( 1 + 2.67T + 4T^{2} \)
3 \( 1 - 2.89T + 9T^{2} \)
5 \( 1 - 3.34iT - 25T^{2} \)
11 \( 1 - 15.3iT - 121T^{2} \)
13 \( 1 + 7.80T + 169T^{2} \)
17 \( 1 + 8.48T + 289T^{2} \)
19 \( 1 + 0.826T + 361T^{2} \)
23 \( 1 + 2.74T + 529T^{2} \)
29 \( 1 + 15.8iT - 841T^{2} \)
31 \( 1 - 52.1iT - 961T^{2} \)
37 \( 1 + 39.7T + 1.36e3T^{2} \)
43 \( 1 + 71.9T + 1.84e3T^{2} \)
47 \( 1 - 0.993T + 2.20e3T^{2} \)
53 \( 1 + 10.7iT - 2.80e3T^{2} \)
59 \( 1 - 24.7iT - 3.48e3T^{2} \)
61 \( 1 - 59.3iT - 3.72e3T^{2} \)
67 \( 1 - 3.55iT - 4.48e3T^{2} \)
71 \( 1 - 49.0iT - 5.04e3T^{2} \)
73 \( 1 - 106. iT - 5.32e3T^{2} \)
79 \( 1 + 109. iT - 6.24e3T^{2} \)
83 \( 1 + 117. iT - 6.88e3T^{2} \)
89 \( 1 - 171.T + 7.92e3T^{2} \)
97 \( 1 + 41.4T + 9.40e3T^{2} \)
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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

−11.72417190660628637399903659877, −10.39882836811437321639917061290, −10.08834211780429188370412621988, −9.148086005908635936964390929973, −8.300313207475055570948313796699, −7.20640185054433145336850993724, −6.86272202209769765946268481668, −4.63427706174880008772259076983, −3.17920977249955730092665278070, −1.90385133757200241360289294318, 0.24672473738149675538382616399, 2.11346350569432105632717204910, 3.38538684073497158123477939407, 5.17018464108266500306712248742, 6.52814251346326576051116588960, 7.912909266540294726985987794059, 8.609550413880150200435396701874, 9.034877882053886074531898995354, 9.808228385817754286843406217855, 10.98564922842053411794641580515

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