↑

Properties

Label 2-287-7.4-c1-0-13
Degree $2$
Conductor $287$
Sign $0.837 - 0.545i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.486 + 0.841i)2-s + (−0.753 + 1.30i)3-s + (0.527 − 0.913i)4-s + (−1.41 − 2.45i)5-s − 1.46·6-s + (2.62 − 0.308i)7-s + 2.96·8-s + (0.365 + 0.632i)9-s + (1.37 − 2.38i)10-s + (−1.68 + 2.92i)11-s + (0.794 + 1.37i)12-s + 6.81·13-s + (1.53 + 2.06i)14-s + 4.27·15-s + (0.388 + 0.672i)16-s + (−0.165 + 0.287i)17-s + ⋯
L(s)  = 1  + (0.343 + 0.595i)2-s + (−0.434 + 0.753i)3-s + (0.263 − 0.456i)4-s + (−0.634 − 1.09i)5-s − 0.597·6-s + (0.993 − 0.116i)7-s + 1.04·8-s + (0.121 + 0.210i)9-s + (0.435 − 0.755i)10-s + (−0.508 + 0.880i)11-s + (0.229 + 0.397i)12-s + 1.88·13-s + (0.410 + 0.551i)14-s + 1.10·15-s + (0.0970 + 0.168i)16-s + (−0.0402 + 0.0696i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.837 - 0.545i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.837 - 0.545i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.837 - 0.545i$
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: \((2,\ 287,\ (\ :1/2),\ 0.837 - 0.545i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.47388 + 0.437615i\)
\(L(\frac12)\) \(\approx\) \(1.47388 + 0.437615i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-2.62 + 0.308i)T \)
41 \( 1 - T \)
good2 \( 1 + (-0.486 - 0.841i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (0.753 - 1.30i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.41 + 2.45i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.68 - 2.92i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 6.81T + 13T^{2} \)
17 \( 1 + (0.165 - 0.287i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.15 + 7.19i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.83 - 4.91i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 1.34T + 29T^{2} \)
31 \( 1 + (0.270 - 0.469i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.10 - 3.64i)T + (-18.5 + 32.0i)T^{2} \)
43 \( 1 + 11.7T + 43T^{2} \)
47 \( 1 + (2.73 + 4.73i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.03 + 3.53i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (7.61 - 13.1i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.360 - 0.624i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.13 + 3.69i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 3.70T + 71T^{2} \)
73 \( 1 + (5.74 - 9.95i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.23 + 7.33i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 7.88T + 83T^{2} \)
89 \( 1 + (-1.15 - 2.00i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 13.3T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.54779038029248254463353735310, −11.07377692599445208747009702187, −10.24232681971968843644682162802, −8.893315916448405474716012362054, −8.034940984118653782947054442142, −6.96296385564259444759367016673, −5.53755152791307461365461478923, −4.78895151935552176039330978422, −4.24309974729289629461215200443, −1.50875559415254585224262113703, 1.62302280066699948740839561713, 3.20804144878694680768156725372, 4.09666415012680670917342123999, 5.94158995959982321151700863446, 6.78613398542337921303900596847, 7.908917526307094743045372384389, 8.413318316763494195944491283546, 10.59715309487632142288468994547, 11.03116532648702802258795448517, 11.57876891611951578075810137701

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