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Properties

Label 2-287-7.2-c1-0-0
Degree $2$
Conductor $287$
Sign $-0.965 + 0.260i$
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  + (−1.36 + 2.36i)2-s + (−1.26 − 2.18i)3-s + (−2.74 − 4.74i)4-s + (0.813 − 1.40i)5-s + 6.89·6-s + (−1.98 + 1.75i)7-s + 9.52·8-s + (−1.68 + 2.91i)9-s + (2.22 + 3.85i)10-s + (0.608 + 1.05i)11-s + (−6.91 + 11.9i)12-s − 3.49·13-s + (−1.43 − 7.09i)14-s − 4.10·15-s + (−7.53 + 13.0i)16-s + (−1.79 − 3.11i)17-s + ⋯
L(s)  = 1  + (−0.967 + 1.67i)2-s + (−0.728 − 1.26i)3-s + (−1.37 − 2.37i)4-s + (0.363 − 0.630i)5-s + 2.81·6-s + (−0.749 + 0.661i)7-s + 3.36·8-s + (−0.560 + 0.970i)9-s + (0.703 + 1.21i)10-s + (0.183 + 0.317i)11-s + (−1.99 + 3.45i)12-s − 0.969·13-s + (−0.383 − 1.89i)14-s − 1.06·15-s + (−1.88 + 3.26i)16-s + (−0.435 − 0.754i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 + 0.260i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.965 + 0.260i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $-0.965 + 0.260i$
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: \((2,\ 287,\ (\ :1/2),\ -0.965 + 0.260i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00776586 - 0.0586771i\)
\(L(\frac12)\) \(\approx\) \(0.00776586 - 0.0586771i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (1.98 - 1.75i)T \)
41 \( 1 - T \)
good2 \( 1 + (1.36 - 2.36i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (1.26 + 2.18i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.813 + 1.40i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.608 - 1.05i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 3.49T + 13T^{2} \)
17 \( 1 + (1.79 + 3.11i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.449 + 0.777i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.23 - 7.34i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 7.30T + 29T^{2} \)
31 \( 1 + (-5.02 - 8.70i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.173 - 0.299i)T + (-18.5 - 32.0i)T^{2} \)
43 \( 1 + 6.77T + 43T^{2} \)
47 \( 1 + (0.0977 - 0.169i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.72 + 4.71i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.31 + 7.47i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.0722 - 0.125i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.29 - 2.24i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 7.50T + 71T^{2} \)
73 \( 1 + (-2.97 - 5.15i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.71 - 11.6i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 10.6T + 83T^{2} \)
89 \( 1 + (-2.88 + 4.99i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 3.67T + 97T^{2} \)
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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

−12.52844197720620443982307370987, −11.45119078994529928658646837242, −9.792625148972828322172003697993, −9.379041763738621069556419518164, −8.262152024144151847613599099561, −7.20206980575507537204749360483, −6.73211464025866626374140030721, −5.66406024974560724073895194874, −5.10032610754748259623560520745, −1.62211487890718196270479145323, 0.06600588509432662924975332202, 2.47325109648297214238141921279, 3.75288059156890137534277251053, 4.49804463007163238652378526161, 6.30617349188070652715970851343, 7.79826901036362243324472742265, 9.115541293599666924609371518430, 9.937776564383589894409901700776, 10.31354744000883810773118048291, 10.92716238583466553746961034775

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