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Properties

Label 2-287-287.286-c2-0-2
Degree $2$
Conductor $287$
Sign $-0.137 - 0.990i$
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-s − 0.913·3-s − 3·4-s − 7.47i·5-s + 0.913·6-s + (−6.10 − 3.41i)7-s + 7·8-s − 8.16·9-s + 7.47i·10-s + 6.11i·11-s + 2.74·12-s + 0.913·13-s + (6.10 + 3.41i)14-s + 6.83i·15-s + 5·16-s − 1.10·17-s + ⋯
L(s)  = 1  − 0.5·2-s − 0.304·3-s − 0.750·4-s − 1.49i·5-s + 0.152·6-s + (−0.872 − 0.488i)7-s + 0.875·8-s − 0.907·9-s + 0.747i·10-s + 0.556i·11-s + 0.228·12-s + 0.0702·13-s + (0.436 + 0.244i)14-s + 0.455i·15-s + 0.312·16-s − 0.0649·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.137 - 0.990i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.137 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $-0.137 - 0.990i$
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: \((2,\ 287,\ (\ :1),\ -0.137 - 0.990i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.107880 + 0.123838i\)
\(L(\frac12)\) \(\approx\) \(0.107880 + 0.123838i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (6.10 + 3.41i)T \)
41 \( 1 + (24.7 + 32.7i)T \)
good2 \( 1 + T + 4T^{2} \)
3 \( 1 + 0.913T + 9T^{2} \)
5 \( 1 + 7.47iT - 25T^{2} \)
11 \( 1 - 6.11iT - 121T^{2} \)
13 \( 1 - 0.913T + 169T^{2} \)
17 \( 1 + 1.10T + 289T^{2} \)
19 \( 1 - 28.6T + 361T^{2} \)
23 \( 1 + 26.7T + 529T^{2} \)
29 \( 1 - 52.5iT - 841T^{2} \)
31 \( 1 - 26.7iT - 961T^{2} \)
37 \( 1 + 18.8T + 1.36e3T^{2} \)
43 \( 1 + 19.4T + 1.84e3T^{2} \)
47 \( 1 - 48.5T + 2.20e3T^{2} \)
53 \( 1 - 44.2iT - 2.80e3T^{2} \)
59 \( 1 + 19.3iT - 3.48e3T^{2} \)
61 \( 1 + 28.0iT - 3.72e3T^{2} \)
67 \( 1 - 14.3iT - 4.48e3T^{2} \)
71 \( 1 + 134. iT - 5.04e3T^{2} \)
73 \( 1 - 44.8iT - 5.32e3T^{2} \)
79 \( 1 - 70.8iT - 6.24e3T^{2} \)
83 \( 1 - 90.3iT - 6.88e3T^{2} \)
89 \( 1 + 81.5T + 7.92e3T^{2} \)
97 \( 1 + 100.T + 9.40e3T^{2} \)
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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

−12.17339924138175691122184596009, −10.68054075133821921624723978126, −9.725475560435023563630712684202, −9.046717122735575716009313568880, −8.308738680758846463273337061254, −7.15323292893863570348344099246, −5.57746521649791053068753460560, −4.86477260945422039501213060429, −3.58966098432155336803908939526, −1.17109606659097100458903843252, 0.11813396438650101172307242186, 2.70569860925481451892827348297, 3.73816441637883516143242842458, 5.57571642339936475068005659461, 6.29051239562938498222104626059, 7.53374325354848808857463678682, 8.507881413900769345621072548970, 9.665301097963690419882422880985, 10.17564761144022623203229845309, 11.31185569444259884028878049188

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