Properties

Label 2-287-7.4-c1-0-0
Degree $2$
Conductor $287$
Sign $-0.538 + 0.842i$
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  + (0.992 + 1.71i)2-s + (−1.44 + 2.50i)3-s + (−0.971 + 1.68i)4-s + (−1.95 − 3.38i)5-s − 5.73·6-s + (−2.26 + 1.36i)7-s + 0.114·8-s + (−2.66 − 4.62i)9-s + (3.88 − 6.73i)10-s + (−1.59 + 2.76i)11-s + (−2.80 − 4.85i)12-s − 4.29·13-s + (−4.59 − 2.54i)14-s + 11.3·15-s + (2.05 + 3.56i)16-s + (−1.05 + 1.83i)17-s + ⋯
L(s)  = 1  + (0.701 + 1.21i)2-s + (−0.833 + 1.44i)3-s + (−0.485 + 0.841i)4-s + (−0.875 − 1.51i)5-s − 2.34·6-s + (−0.856 + 0.516i)7-s + 0.0404·8-s + (−0.889 − 1.54i)9-s + (1.22 − 2.12i)10-s + (−0.481 + 0.834i)11-s + (−0.809 − 1.40i)12-s − 1.19·13-s + (−1.22 − 0.679i)14-s + 2.91·15-s + (0.514 + 0.890i)16-s + (−0.256 + 0.444i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.538 + 0.842i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.538 + 0.842i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $-0.538 + 0.842i$
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: \((2,\ 287,\ (\ :1/2),\ -0.538 + 0.842i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.309542 - 0.564944i\)
\(L(\frac12)\) \(\approx\) \(0.309542 - 0.564944i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (2.26 - 1.36i)T \)
41 \( 1 - T \)
good2 \( 1 + (-0.992 - 1.71i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (1.44 - 2.50i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.95 + 3.38i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.59 - 2.76i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 4.29T + 13T^{2} \)
17 \( 1 + (1.05 - 1.83i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.26 - 5.66i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.457 + 0.791i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 5.76T + 29T^{2} \)
31 \( 1 + (-0.983 + 1.70i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.267 - 0.464i)T + (-18.5 + 32.0i)T^{2} \)
43 \( 1 + 4.90T + 43T^{2} \)
47 \( 1 + (2.79 + 4.84i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.18 - 2.05i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.11 - 5.39i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.09 - 1.89i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.62 - 9.73i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 3.34T + 71T^{2} \)
73 \( 1 + (-0.883 + 1.52i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.73 + 9.93i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 16.3T + 83T^{2} \)
89 \( 1 + (-5.55 - 9.62i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 14.5T + 97T^{2} \)
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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

−12.28253033281924399867973682114, −11.96419723461592970744216684027, −10.30265731676132440304427773862, −9.656628362039139768929021181602, −8.538467776005248168521049043910, −7.49818628417967579082543822720, −6.09691966075263176587814127370, −5.16633760312364436108075838361, −4.70236324500952386742083100755, −3.79238307602226299030826824233, 0.41109828042779016768890521282, 2.59812359561370659674326694592, 3.22911018991727554798800759836, 4.89739471437921662387587940136, 6.40934967302751981209532236173, 7.14046387535075694791801666201, 7.75066891528971830725435732242, 9.920019151453014750698173926628, 10.86787811583130283080772761187, 11.35500324794480737909293766140

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