Properties

Label 2-287-287.286-c2-0-10
Degree $2$
Conductor $287$
Sign $-0.249 - 0.968i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.249 - 0.968i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.249 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $-0.249 - 0.968i$
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: \((2,\ 287,\ (\ :1),\ -0.249 - 0.968i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.263951 + 0.340457i\)
\(L(\frac12)\) \(\approx\) \(0.263951 + 0.340457i\)
\(L(2)\) 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

−11.32778307855292670231284669416, −10.96685804401013697516519509469, −10.17539751112338287178462317841, −8.698843874925229469916725252677, −8.472769648164210626083451366783, −7.05163240866944319730128355663, −6.05267966958544152388805681495, −5.10391700354190514361048729801, −3.06939290789486192477964732520, −1.20480319439832146873323310580, 0.45471135450581538263085411534, 1.67413689427955200168937267822, 4.30447422478110096998866047116, 5.20331178737960668367634330872, 6.65155069245738258127321990012, 7.63938529362682197257411031433, 8.438569941261342772494572848516, 9.546201594260614472684506205986, 10.30780821531997052502157213915, 11.11268586485919010260502964904

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