↑

Properties

Label 2-1950-65.9-c1-0-4
Degree $2$
Conductor $1950$
Sign $-0.892 - 0.450i$
Analytic cond. $15.5708$
Root an. cond. $3.94598$
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  + (0.866 + 0.5i)2-s + (−0.866 − 0.5i)3-s + (0.499 + 0.866i)4-s + (−0.499 − 0.866i)6-s + (0.486 − 0.280i)7-s + 0.999i·8-s + (0.499 + 0.866i)9-s + (−2.06 + 3.57i)11-s − 0.999i·12-s + (−2.21 + 2.84i)13-s + 0.561·14-s + (−0.5 + 0.866i)16-s + (2.70 − 1.56i)17-s + 0.999i·18-s + (0.280 + 0.486i)19-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (−0.499 − 0.288i)3-s + (0.249 + 0.433i)4-s + (−0.204 − 0.353i)6-s + (0.183 − 0.106i)7-s + 0.353i·8-s + (0.166 + 0.288i)9-s + (−0.621 + 1.07i)11-s − 0.288i·12-s + (−0.615 + 0.788i)13-s + 0.150·14-s + (−0.125 + 0.216i)16-s + (0.655 − 0.378i)17-s + 0.235i·18-s + (0.0644 + 0.111i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.892 - 0.450i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.892 - 0.450i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $-0.892 - 0.450i$
Analytic conductor: \(15.5708\)
Root analytic conductor: \(3.94598\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1950} (1699, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1950,\ (\ :1/2),\ -0.892 - 0.450i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.016927979\)
\(L(\frac12)\) \(\approx\) \(1.016927979\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 + (0.866 + 0.5i)T \)
5 \( 1 \)
13 \( 1 + (2.21 - 2.84i)T \)
good7 \( 1 + (-0.486 + 0.280i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.06 - 3.57i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-2.70 + 1.56i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.280 - 0.486i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.05 + 2.34i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.21 + 2.11i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 6.68T + 31T^{2} \)
37 \( 1 + (3.57 + 2.06i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (6.12 - 10.6i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.379 + 0.219i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 7iT - 47T^{2} \)
53 \( 1 - 8.56iT - 53T^{2} \)
59 \( 1 + (-3.21 - 5.57i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3 + 5.19i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.94 + 1.12i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-6.56 - 11.3i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 9.36iT - 73T^{2} \)
79 \( 1 + 11.5T + 79T^{2} \)
83 \( 1 + 7.12iT - 83T^{2} \)
89 \( 1 + (9.40 - 16.2i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-6.16 + 3.56i)T + (48.5 - 84.0i)T^{2} \)
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

−9.696220471171243372829604035182, −8.531168881173127057329130935739, −7.60465173441841829292435482720, −7.18845588201989688078632088124, −6.35416544788028519188191623704, −5.42543655107893714342193747147, −4.77686419466639840357968786108, −4.01528007078061724163908336719, −2.67084618996389664672651651141, −1.70044406178999949016673509209, 0.29714564399343292272137790308, 1.82172302761490790217750419846, 3.12854865285485686396537603424, 3.72622862256365874806523652855, 5.01397329577790064024738258177, 5.45001205864214459253627647655, 6.13589401744733338545118661824, 7.23074999489066486069284736949, 8.040139832995154365125390050068, 8.895167985079832529276491953559

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