↑

Properties

Label 2-950-19.11-c1-0-27
Degree $2$
Conductor $950$
Sign $-0.634 + 0.772i$
Analytic cond. $7.58578$
Root an. cond. $2.75423$
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  + (−0.5 + 0.866i)2-s + (0.341 − 0.590i)3-s + (−0.499 − 0.866i)4-s + (0.341 + 0.590i)6-s − 0.317·7-s + 0.999·8-s + (1.26 + 2.19i)9-s − 4.31·11-s − 0.682·12-s + (−3.14 − 5.45i)13-s + (0.158 − 0.275i)14-s + (−0.5 + 0.866i)16-s + (0.0669 − 0.115i)17-s − 2.53·18-s + (−4.05 − 1.60i)19-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.196 − 0.341i)3-s + (−0.249 − 0.433i)4-s + (0.139 + 0.241i)6-s − 0.120·7-s + 0.353·8-s + (0.422 + 0.731i)9-s − 1.29·11-s − 0.196·12-s + (−0.873 − 1.51i)13-s + (0.0424 − 0.0735i)14-s + (−0.125 + 0.216i)16-s + (0.0162 − 0.0281i)17-s − 0.597·18-s + (−0.929 − 0.367i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.634 + 0.772i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.634 + 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.634 + 0.772i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (201, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.634 + 0.772i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.140552 - 0.297299i\)
\(L(\frac12)\) \(\approx\) \(0.140552 - 0.297299i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 \)
19 \( 1 + (4.05 + 1.60i)T \)
good3 \( 1 + (-0.341 + 0.590i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + 0.317T + 7T^{2} \)
11 \( 1 + 4.31T + 11T^{2} \)
13 \( 1 + (3.14 + 5.45i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.0669 + 0.115i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (1.98 + 3.44i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.57 - 7.92i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 2.98T + 31T^{2} \)
37 \( 1 + 5.07T + 37T^{2} \)
41 \( 1 + (-0.433 + 0.750i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.85 - 4.93i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (6.48 + 11.2i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.96 + 6.86i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.80 + 8.32i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.08 + 5.34i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.295 - 0.511i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (5.83 - 10.0i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-4.13 + 7.15i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.66 + 2.87i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 4.20T + 83T^{2} \)
89 \( 1 + (-1.85 - 3.22i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.42 - 4.20i)T + (-48.5 - 84.0i)T^{2} \)
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

−9.922798903916987538351296848998, −8.470201272741297006271829096682, −8.139592628966189906898101456115, −7.30310028343870608233484540228, −6.54901628821430004032311172176, −5.21147204726963166683233702870, −4.89892850274159833407510705487, −3.12284101882720555449499487054, −2.04731665134523931866722633506, −0.15649717172150991007035575650, 1.81876262299425626992108586709, 2.88181792382334428435581839427, 4.06553587797126204129162504379, 4.74196730598372758272984817831, 6.10813005071394217586285696486, 7.10578872476106693305261152465, 7.975796968143142447168750005183, 8.885127791478456243387638400257, 9.698970111736579651936416675460, 10.11520722473471453221698438522

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