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Properties

Label 2-287-7.4-c1-0-7
Degree $2$
Conductor $287$
Sign $-0.711 - 0.702i$
Analytic cond. $2.29170$
Root an. cond. $1.51383$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.889 + 1.54i)2-s + (−0.603 + 1.04i)3-s + (−0.582 + 1.00i)4-s + (1.26 + 2.19i)5-s − 2.14·6-s + (−0.862 − 2.50i)7-s + 1.48·8-s + (0.770 + 1.33i)9-s + (−2.25 + 3.90i)10-s + (−2.34 + 4.06i)11-s + (−0.703 − 1.21i)12-s − 2.97·13-s + (3.08 − 3.55i)14-s − 3.05·15-s + (2.48 + 4.30i)16-s + (3.50 − 6.07i)17-s + ⋯
L(s)  = 1  + (0.628 + 1.08i)2-s + (−0.348 + 0.603i)3-s + (−0.291 + 0.504i)4-s + (0.566 + 0.980i)5-s − 0.877·6-s + (−0.325 − 0.945i)7-s + 0.525·8-s + (0.256 + 0.444i)9-s + (−0.712 + 1.23i)10-s + (−0.707 + 1.22i)11-s + (−0.203 − 0.351i)12-s − 0.825·13-s + (0.824 − 0.949i)14-s − 0.789·15-s + (0.621 + 1.07i)16-s + (0.850 − 1.47i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.711 - 0.702i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.711 - 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $-0.711 - 0.702i$
Analytic conductor: \(2.29170\)
Root analytic conductor: \(1.51383\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (165, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 287,\ (\ :1/2),\ -0.711 - 0.702i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.650364 + 1.58521i\)
\(L(\frac12)\) \(\approx\) \(0.650364 + 1.58521i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.862 + 2.50i)T \)
41 \( 1 - T \)
good2 \( 1 + (-0.889 - 1.54i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (0.603 - 1.04i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.26 - 2.19i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.34 - 4.06i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 2.97T + 13T^{2} \)
17 \( 1 + (-3.50 + 6.07i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.39 + 2.41i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.02 + 3.50i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 7.66T + 29T^{2} \)
31 \( 1 + (0.621 - 1.07i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.74 - 4.74i)T + (-18.5 + 32.0i)T^{2} \)
43 \( 1 - 9.18T + 43T^{2} \)
47 \( 1 + (-3.88 - 6.72i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.72 + 8.17i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.816 - 1.41i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.97 + 12.0i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.87 + 3.25i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 6.77T + 71T^{2} \)
73 \( 1 + (1.35 - 2.34i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.23 + 10.8i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 0.292T + 83T^{2} \)
89 \( 1 + (1.49 + 2.59i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 11.1T + 97T^{2} \)
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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

−12.40682697940004743997821746230, −10.89122231073931509108146996634, −10.19027607780765988643831662951, −9.810463179652971033570728037268, −7.64554512641349472958347545984, −7.20333185085516933601843194894, −6.30971749117835409409629146864, −4.97515180502734013532068872249, −4.49679303439757848893012326559, −2.61920185680496829191750842876, 1.27263045360449444380384491321, 2.60075906346797882403464941864, 3.96999331067658679171247495648, 5.49855309167899563793185108703, 5.94418112295429274387440334762, 7.67422690385488140621899745318, 8.727421075162812180612347982633, 9.824334201595797552514853403239, 10.69084088523448437627506877242, 11.99824328903009796455257293789

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