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Properties

Label 2-287-7.2-c1-0-4
Degree $2$
Conductor $287$
Sign $0.605 - 0.795i$
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  + (−1.15 − 1.99i)3-s + (1 + 1.73i)4-s + (−0.651 + 1.12i)5-s + (−0.5 + 2.59i)7-s + (−1.15 + 1.99i)9-s + (2.15 + 3.72i)11-s + (2.30 − 3.98i)12-s − 2.30·13-s + 2.99·15-s + (−1.99 + 3.46i)16-s + (1.95 + 3.38i)17-s + (1.80 − 3.12i)19-s − 2.60·20-s + (5.75 − 1.99i)21-s + (2.80 − 4.85i)23-s + ⋯
L(s)  = 1  + (−0.664 − 1.15i)3-s + (0.5 + 0.866i)4-s + (−0.291 + 0.504i)5-s + (−0.188 + 0.981i)7-s + (−0.383 + 0.664i)9-s + (0.648 + 1.12i)11-s + (0.664 − 1.15i)12-s − 0.638·13-s + 0.774·15-s + (−0.499 + 0.866i)16-s + (0.473 + 0.820i)17-s + (0.413 − 0.716i)19-s − 0.582·20-s + (1.25 − 0.435i)21-s + (0.584 − 1.01i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.605 - 0.795i$
Analytic conductor: \(2.29170\)
Root analytic conductor: \(1.51383\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (247, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 287,\ (\ :1/2),\ 0.605 - 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.918511 + 0.455331i\)
\(L(\frac12)\) \(\approx\) \(0.918511 + 0.455331i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.5 - 2.59i)T \)
41 \( 1 + T \)
good2 \( 1 + (-1 - 1.73i)T^{2} \)
3 \( 1 + (1.15 + 1.99i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (0.651 - 1.12i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.15 - 3.72i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 2.30T + 13T^{2} \)
17 \( 1 + (-1.95 - 3.38i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.80 + 3.12i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.80 + 4.85i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 5.60T + 29T^{2} \)
31 \( 1 + (-1.15 - 1.99i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.60 - 6.24i)T + (-18.5 - 32.0i)T^{2} \)
43 \( 1 + 9.60T + 43T^{2} \)
47 \( 1 + (-4.5 + 7.79i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.95 + 8.58i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.197 + 0.341i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.60 - 6.24i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.30 - 5.72i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 10.8T + 71T^{2} \)
73 \( 1 + (4.65 + 8.05i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-8.71 + 15.0i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 11.6T + 83T^{2} \)
89 \( 1 + (0.197 - 0.341i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 9.30T + 97T^{2} \)
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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

−12.07169130891618149726245898176, −11.52527880174196381288264817971, −10.25874406966632061929562525186, −8.881345045608540317746609932038, −7.83415839096609689563759328485, −6.76832473440728236884229326331, −6.61898200005679168529160554357, −4.96270962690721224131353462943, −3.19689118988013706878348158487, −1.95061106378245794966864197889, 0.871327471241344648048199442923, 3.41871702756436775062776313926, 4.64965001231141741102183587495, 5.44787365453602291962958762837, 6.53204927104804491202499802319, 7.72989030003612269256334314418, 9.278167188241163486260069375504, 9.921663802364087070787711847278, 10.74374735448137007906916407422, 11.39977628597823452827991904386

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