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Properties

Label 2-287-7.4-c1-0-1
Degree $2$
Conductor $287$
Sign $0.386 - 0.922i$
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  + (−0.809 − 1.40i)2-s + (−0.5 + 0.866i)3-s + (−0.309 + 0.535i)4-s + (0.5 + 0.866i)5-s + 1.61·6-s + (−2 + 1.73i)7-s − 2.23·8-s + (1 + 1.73i)9-s + (0.809 − 1.40i)10-s + (−0.5 + 0.866i)11-s + (−0.309 − 0.535i)12-s − 4.47·13-s + (4.04 + 1.40i)14-s − 0.999·15-s + (2.42 + 4.20i)16-s + (−2.11 + 3.66i)17-s + ⋯
L(s)  = 1  + (−0.572 − 0.990i)2-s + (−0.288 + 0.499i)3-s + (−0.154 + 0.267i)4-s + (0.223 + 0.387i)5-s + 0.660·6-s + (−0.755 + 0.654i)7-s − 0.790·8-s + (0.333 + 0.577i)9-s + (0.255 − 0.443i)10-s + (−0.150 + 0.261i)11-s + (−0.0892 − 0.154i)12-s − 1.24·13-s + (1.08 + 0.374i)14-s − 0.258·15-s + (0.606 + 1.05i)16-s + (−0.513 + 0.889i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.386 - 0.922i$
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: \((2,\ 287,\ (\ :1/2),\ 0.386 - 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.462242 + 0.307475i\)
\(L(\frac12)\) \(\approx\) \(0.462242 + 0.307475i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (2 - 1.73i)T \)
41 \( 1 + T \)
good2 \( 1 + (0.809 + 1.40i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (0.5 - 0.866i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.5 - 0.866i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 4.47T + 13T^{2} \)
17 \( 1 + (2.11 - 3.66i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.736 + 1.27i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.35 - 7.54i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 4.47T + 29T^{2} \)
31 \( 1 + (-0.118 + 0.204i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.73 - 3.00i)T + (-18.5 + 32.0i)T^{2} \)
43 \( 1 + 6.47T + 43T^{2} \)
47 \( 1 + (-1.73 - 3.00i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.11 + 8.86i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.35 - 7.54i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.73 + 8.20i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.97 - 5.14i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 14.4T + 71T^{2} \)
73 \( 1 + (-1.26 + 2.18i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 1.52T + 83T^{2} \)
89 \( 1 + (-7.11 - 12.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 8.47T + 97T^{2} \)
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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

−11.77872559265066514841536305596, −10.86371986813986674615591661868, −10.07804717584970103862343828416, −9.643298441747539081339442801658, −8.580423976387687717036542409587, −7.09735130777205936866009321661, −5.96609076022523136848228535054, −4.78397126170656020804853730748, −3.11995816372698908113105286069, −2.07373918354998676449009635912, 0.49220371385718397235918026160, 2.94644447018086974448287859589, 4.71520534815571736023667794479, 6.08885885001394570420387556522, 6.93322798760280802768147733316, 7.40398738329075610382587225464, 8.762044341983082142054497535317, 9.465160227440721298044699560769, 10.44629171692350751491236166204, 11.89663849528871112560557420888

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