↑

Properties

Label 2-287-41.37-c1-0-13
Degree $2$
Conductor $287$
Sign $0.956 - 0.292i$
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  + (0.178 − 0.129i)2-s + 1.96·3-s + (−0.602 + 1.85i)4-s + (0.495 − 1.52i)5-s + (0.351 − 0.255i)6-s + (0.809 + 0.587i)7-s + (0.269 + 0.829i)8-s + 0.865·9-s + (−0.109 − 0.336i)10-s + (1.10 + 3.39i)11-s + (−1.18 + 3.64i)12-s + (3.74 − 2.71i)13-s + 0.220·14-s + (0.973 − 2.99i)15-s + (−3.00 − 2.18i)16-s + (−1.09 − 3.38i)17-s + ⋯
L(s)  = 1  + (0.126 − 0.0917i)2-s + 1.13·3-s + (−0.301 + 0.927i)4-s + (0.221 − 0.681i)5-s + (0.143 − 0.104i)6-s + (0.305 + 0.222i)7-s + (0.0953 + 0.293i)8-s + 0.288·9-s + (−0.0345 − 0.106i)10-s + (0.332 + 1.02i)11-s + (−0.342 + 1.05i)12-s + (1.03 − 0.753i)13-s + 0.0590·14-s + (0.251 − 0.773i)15-s + (−0.750 − 0.545i)16-s + (−0.266 − 0.820i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 - 0.292i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.956 - 0.292i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.956 - 0.292i$
Analytic conductor: \(2.29170\)
Root analytic conductor: \(1.51383\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (78, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 287,\ (\ :1/2),\ 0.956 - 0.292i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.87351 + 0.279821i\)
\(L(\frac12)\) \(\approx\) \(1.87351 + 0.279821i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.69 + 5.23i)T \)
good2 \( 1 + (-0.178 + 0.129i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 - 1.96T + 3T^{2} \)
5 \( 1 + (-0.495 + 1.52i)T + (-4.04 - 2.93i)T^{2} \)
11 \( 1 + (-1.10 - 3.39i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (-3.74 + 2.71i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (1.09 + 3.38i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-0.931 - 0.676i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (6.11 - 4.44i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (0.755 - 2.32i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (2.63 + 8.11i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-3.37 + 10.3i)T + (-29.9 - 21.7i)T^{2} \)
43 \( 1 + (4.23 - 3.07i)T + (13.2 - 40.8i)T^{2} \)
47 \( 1 + (2.41 - 1.75i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (2.46 - 7.59i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (5.76 - 4.19i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (9.61 + 6.98i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (0.0768 - 0.236i)T + (-54.2 - 39.3i)T^{2} \)
71 \( 1 + (2.30 + 7.09i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 - 6.37T + 73T^{2} \)
79 \( 1 - 4.03T + 79T^{2} \)
83 \( 1 + 10.7T + 83T^{2} \)
89 \( 1 + (-3.07 - 2.23i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (4.52 - 13.9i)T + (-78.4 - 57.0i)T^{2} \)
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

−12.10041960983503096592545018793, −11.07939805582020428550471954083, −9.352021358291696422361618773383, −9.165672958329205305164441546785, −7.996525521892966940475171689087, −7.52851112729050110993739734828, −5.68738283761673353309576918053, −4.37685238082501599938535117011, −3.38675567819194111963072784986, −2.05964437259910700183825073716, 1.70121217271526060587438244409, 3.24809597372865738055884888433, 4.37503423991369902148286570496, 6.01260893157253196084126868286, 6.63611310457792144364271848785, 8.294007474002547624709880328757, 8.745707307429971596529138597172, 9.871599831490917824828659642268, 10.73241292729080713557341360769, 11.50690937679745143814356173685

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