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Properties

Label 2-287-287.286-c2-0-50
Degree $2$
Conductor $287$
Sign $-0.945 + 0.324i$
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  + 2.70·2-s − 3.78·3-s + 3.33·4-s − 4.10i·5-s − 10.2·6-s + (−0.503 + 6.98i)7-s − 1.79·8-s + 5.28·9-s − 11.1i·10-s − 16.1i·11-s − 12.6·12-s − 19.5·13-s + (−1.36 + 18.9i)14-s + 15.5i·15-s − 18.2·16-s − 0.467·17-s + ⋯
L(s)  = 1  + 1.35·2-s − 1.26·3-s + 0.834·4-s − 0.820i·5-s − 1.70·6-s + (−0.0719 + 0.997i)7-s − 0.224·8-s + 0.587·9-s − 1.11i·10-s − 1.47i·11-s − 1.05·12-s − 1.50·13-s + (−0.0975 + 1.35i)14-s + 1.03i·15-s − 1.13·16-s − 0.0275·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.945 + 0.324i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.945 + 0.324i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $-0.945 + 0.324i$
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: \((2,\ 287,\ (\ :1),\ -0.945 + 0.324i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0878050 - 0.525737i\)
\(L(\frac12)\) \(\approx\) \(0.0878050 - 0.525737i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.503 - 6.98i)T \)
41 \( 1 + (16.0 + 37.7i)T \)
good2 \( 1 - 2.70T + 4T^{2} \)
3 \( 1 + 3.78T + 9T^{2} \)
5 \( 1 + 4.10iT - 25T^{2} \)
11 \( 1 + 16.1iT - 121T^{2} \)
13 \( 1 + 19.5T + 169T^{2} \)
17 \( 1 + 0.467T + 289T^{2} \)
19 \( 1 + 6.39T + 361T^{2} \)
23 \( 1 + 39.4T + 529T^{2} \)
29 \( 1 + 31.8iT - 841T^{2} \)
31 \( 1 - 44.6iT - 961T^{2} \)
37 \( 1 - 16.9T + 1.36e3T^{2} \)
43 \( 1 - 35.3T + 1.84e3T^{2} \)
47 \( 1 - 76.3T + 2.20e3T^{2} \)
53 \( 1 + 5.38iT - 2.80e3T^{2} \)
59 \( 1 + 96.4iT - 3.48e3T^{2} \)
61 \( 1 - 58.9iT - 3.72e3T^{2} \)
67 \( 1 - 85.1iT - 4.48e3T^{2} \)
71 \( 1 + 62.7iT - 5.04e3T^{2} \)
73 \( 1 + 68.0iT - 5.32e3T^{2} \)
79 \( 1 - 88.4iT - 6.24e3T^{2} \)
83 \( 1 - 16.9iT - 6.88e3T^{2} \)
89 \( 1 + 61.9T + 7.92e3T^{2} \)
97 \( 1 + 19.4T + 9.40e3T^{2} \)
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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

−11.77236944196868695942101764565, −10.64631537525260010372142830827, −9.281945280500948754546883953730, −8.302766843066933697804368246424, −6.56738940567186563340439076871, −5.65758069695186417016195058423, −5.30963851766534707549407255585, −4.22730577759410343165339486197, −2.62015027080049712169032841277, −0.18306413620611056875838884811, 2.49134540163003752989740701848, 4.14151498781191951215138185254, 4.78167992868649119701454543640, 5.91541293503445976817886752993, 6.83244502533693777002139362921, 7.47314507106682957947707670831, 9.695820460667098195166888733311, 10.42200905680437504525016467329, 11.32286850590264671882638942719, 12.24850500019644237776814149770

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