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Properties

Label 2-287-41.10-c1-0-3
Degree $2$
Conductor $287$
Sign $-0.822 - 0.569i$
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  + (2.01 + 1.46i)2-s − 2.38·3-s + (1.30 + 4.02i)4-s + (0.286 + 0.881i)5-s + (−4.81 − 3.49i)6-s + (−0.809 + 0.587i)7-s + (−1.72 + 5.29i)8-s + 2.68·9-s + (−0.714 + 2.19i)10-s + (−1.42 + 4.39i)11-s + (−3.11 − 9.59i)12-s + (−0.706 − 0.513i)13-s − 2.49·14-s + (−0.682 − 2.10i)15-s + (−4.40 + 3.19i)16-s + (0.590 − 1.81i)17-s + ⋯
L(s)  = 1  + (1.42 + 1.03i)2-s − 1.37·3-s + (0.653 + 2.01i)4-s + (0.128 + 0.394i)5-s + (−1.96 − 1.42i)6-s + (−0.305 + 0.222i)7-s + (−0.608 + 1.87i)8-s + 0.894·9-s + (−0.226 + 0.695i)10-s + (−0.430 + 1.32i)11-s + (−0.899 − 2.76i)12-s + (−0.195 − 0.142i)13-s − 0.667·14-s + (−0.176 − 0.542i)15-s + (−1.10 + 0.799i)16-s + (0.143 − 0.440i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.822 - 0.569i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.822 - 0.569i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $-0.822 - 0.569i$
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: \((2,\ 287,\ (\ :1/2),\ -0.822 - 0.569i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.490787 + 1.57173i\)
\(L(\frac12)\) \(\approx\) \(0.490787 + 1.57173i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.92 - 6.10i)T \)
good2 \( 1 + (-2.01 - 1.46i)T + (0.618 + 1.90i)T^{2} \)
3 \( 1 + 2.38T + 3T^{2} \)
5 \( 1 + (-0.286 - 0.881i)T + (-4.04 + 2.93i)T^{2} \)
11 \( 1 + (1.42 - 4.39i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (0.706 + 0.513i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-0.590 + 1.81i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-1.62 + 1.17i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-5.01 - 3.64i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (1.43 + 4.40i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (1.69 - 5.21i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (2.32 + 7.16i)T + (-29.9 + 21.7i)T^{2} \)
43 \( 1 + (-8.53 - 6.19i)T + (13.2 + 40.8i)T^{2} \)
47 \( 1 + (-4.29 - 3.12i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (0.270 + 0.832i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (7.52 + 5.46i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-4.48 + 3.25i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (4.17 + 12.8i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (-1.02 + 3.16i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 - 1.07T + 73T^{2} \)
79 \( 1 + 13.2T + 79T^{2} \)
83 \( 1 + 15.0T + 83T^{2} \)
89 \( 1 + (-13.5 + 9.81i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-1.28 - 3.96i)T + (-78.4 + 57.0i)T^{2} \)
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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

−12.52516131445415580225370256755, −11.60098338433253816379107828271, −10.67705077382312483191490610169, −9.445764152975081552302830229125, −7.58508022056241100841600350775, −6.96770827951879968002842821063, −6.08216176208016122420466851439, −5.21683891400823378661359949366, −4.54924101839979606546133577146, −2.91221232155968857331741579095, 0.986296153983014380713950872523, 2.96891774985584346548692247430, 4.24479660959132190417731335604, 5.41174453967631528503972404248, 5.75220115663548026904612173086, 6.94308329158728909498262211535, 8.837955822445510594246305123295, 10.32572787422315441255547306234, 10.82912280991928500416980183033, 11.52941613875757199156032536802

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