↑

Properties

Label 2-1001-7.4-c1-0-26
Degree $2$
Conductor $1001$
Sign $0.285 + 0.958i$
Analytic cond. $7.99302$
Root an. cond. $2.82719$
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  + (−1.38 − 2.39i)2-s + (−1.49 + 2.58i)3-s + (−2.81 + 4.86i)4-s + (1.63 + 2.83i)5-s + 8.25·6-s + (−2.03 − 1.69i)7-s + 10.0·8-s + (−2.96 − 5.13i)9-s + (4.52 − 7.83i)10-s + (−0.5 + 0.866i)11-s + (−8.40 − 14.5i)12-s − 13-s + (−1.24 + 7.19i)14-s − 9.79·15-s + (−8.18 − 14.1i)16-s + (2.09 − 3.63i)17-s + ⋯
L(s)  = 1  + (−0.976 − 1.69i)2-s + (−0.862 + 1.49i)3-s + (−1.40 + 2.43i)4-s + (0.733 + 1.26i)5-s + 3.36·6-s + (−0.768 − 0.639i)7-s + 3.53·8-s + (−0.988 − 1.71i)9-s + (1.43 − 2.47i)10-s + (−0.150 + 0.261i)11-s + (−2.42 − 4.20i)12-s − 0.277·13-s + (−0.331 + 1.92i)14-s − 2.52·15-s + (−2.04 − 3.54i)16-s + (0.509 − 0.882i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.285 + 0.958i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.285 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1001\)    =    \(7 \cdot 11 \cdot 13\)
Sign: $0.285 + 0.958i$
Analytic conductor: \(7.99302\)
Root analytic conductor: \(2.82719\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1001} (144, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1001,\ (\ :1/2),\ 0.285 + 0.958i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4396548431\)
\(L(\frac12)\) \(\approx\) \(0.4396548431\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (2.03 + 1.69i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + T \)
good2 \( 1 + (1.38 + 2.39i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (1.49 - 2.58i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.63 - 2.83i)T + (-2.5 + 4.33i)T^{2} \)
17 \( 1 + (-2.09 + 3.63i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.87 + 6.70i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.16 - 2.01i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 6.38T + 29T^{2} \)
31 \( 1 + (-2.36 + 4.09i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.960 + 1.66i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 0.0990T + 41T^{2} \)
43 \( 1 + 0.570T + 43T^{2} \)
47 \( 1 + (2.18 + 3.78i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.585 + 1.01i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.25 + 3.90i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.392 - 0.678i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.46 + 4.27i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 3.90T + 71T^{2} \)
73 \( 1 + (4.14 - 7.17i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.77 - 8.27i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 9.49T + 83T^{2} \)
89 \( 1 + (0.146 + 0.254i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 8.10T + 97T^{2} \)
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

−10.03045200704684204375447339078, −9.624717027228601204798097028399, −8.827196318435015400095028439331, −7.33492538650453567862127239041, −6.51288818445377969145833741841, −5.02229831700277109640583491332, −4.15517724451761039157229915083, −3.17560482427372538768778046417, −2.55301151311778370993905106748, −0.42017481326407008801961160144, 0.935374372003877241315406378251, 1.82952075900984576445492798559, 4.77650881144578848029623054572, 5.61376318103288182620134442639, 6.11432730823099514646791443397, 6.53750232775187406543660850626, 7.64651698886499752825837985320, 8.438520083668218983152949066839, 8.772177382937949098223389173304, 9.959071840935014490511533485664

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