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Properties

Label 2-287-41.10-c1-0-19
Degree $2$
Conductor $287$
Sign $-0.767 + 0.640i$
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  + (−0.309 − 0.224i)2-s + 0.381·3-s + (−0.572 − 1.76i)4-s + (−0.5 − 1.53i)5-s + (−0.118 − 0.0857i)6-s + (−0.809 + 0.587i)7-s + (−0.454 + 1.40i)8-s − 2.85·9-s + (−0.190 + 0.587i)10-s + (1.19 − 3.66i)11-s + (−0.218 − 0.673i)12-s + (−1.80 − 1.31i)13-s + 0.381·14-s + (−0.190 − 0.587i)15-s + (−2.54 + 1.84i)16-s + (−0.454 + 1.40i)17-s + ⋯
L(s)  = 1  + (−0.218 − 0.158i)2-s + 0.220·3-s + (−0.286 − 0.881i)4-s + (−0.223 − 0.688i)5-s + (−0.0481 − 0.0350i)6-s + (−0.305 + 0.222i)7-s + (−0.160 + 0.495i)8-s − 0.951·9-s + (−0.0603 + 0.185i)10-s + (0.359 − 1.10i)11-s + (−0.0631 − 0.194i)12-s + (−0.501 − 0.364i)13-s + 0.102·14-s + (−0.0493 − 0.151i)15-s + (−0.636 + 0.462i)16-s + (−0.110 + 0.339i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.767 + 0.640i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.767 + 0.640i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $-0.767 + 0.640i$
Analytic conductor: \(2.29170\)
Root analytic conductor: \(1.51383\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (92, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 287,\ (\ :1/2),\ -0.767 + 0.640i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.259435 - 0.716115i\)
\(L(\frac12)\) \(\approx\) \(0.259435 - 0.716115i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.809 - 0.587i)T \)
41 \( 1 + (-6.39 + 0.224i)T \)
good2 \( 1 + (0.309 + 0.224i)T + (0.618 + 1.90i)T^{2} \)
3 \( 1 - 0.381T + 3T^{2} \)
5 \( 1 + (0.5 + 1.53i)T + (-4.04 + 2.93i)T^{2} \)
11 \( 1 + (-1.19 + 3.66i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (1.80 + 1.31i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (0.454 - 1.40i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (0.618 - 0.449i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-0.190 - 0.138i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (1.23 + 3.80i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-2.69 + 8.28i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (1.66 + 5.11i)T + (-29.9 + 21.7i)T^{2} \)
43 \( 1 + (3.61 + 2.62i)T + (13.2 + 40.8i)T^{2} \)
47 \( 1 + (-9.35 - 6.79i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-3.47 - 10.6i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-4.35 - 3.16i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (2.04 - 1.48i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (1.21 + 3.75i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (-2.71 + 8.36i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 - 10.7T + 73T^{2} \)
79 \( 1 + 12.0T + 79T^{2} \)
83 \( 1 + 6.85T + 83T^{2} \)
89 \( 1 + (-4.11 + 2.99i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-2.97 - 9.14i)T + (-78.4 + 57.0i)T^{2} \)
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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

−11.35707870356257177714561447937, −10.53255841144968789792432958189, −9.330070630150909539743955054742, −8.816532180009945179310423623493, −7.907116031167390262388444913344, −6.12660993267368793682186992784, −5.56075798217903364529095919692, −4.18569648519113190939888819815, −2.56593710766165542748577295262, −0.58790613784017148161211276018, 2.63915257937214057513340106684, 3.66515986688993834627692896188, 4.94653616005319035244306697577, 6.76976257217345352328712391909, 7.19748650295974274576695492693, 8.397170046302914718985172613471, 9.208672598821457507208305144487, 10.17428908475832576613051800456, 11.38846155597043514304768621868, 12.14250002132096884871511003900

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