↑

Properties

Label 2-1950-65.49-c1-0-5
Degree $2$
Conductor $1950$
Sign $-0.999 - 0.0280i$
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  + (0.5 + 0.866i)2-s + (0.866 − 0.5i)3-s + (−0.499 + 0.866i)4-s + (0.866 + 0.499i)6-s + (0.661 − 1.14i)7-s − 0.999·8-s + (0.499 − 0.866i)9-s + (−3.99 + 2.30i)11-s + 0.999i·12-s + (−1.66 + 3.20i)13-s + 1.32·14-s + (−0.5 − 0.866i)16-s + (−3.46 − 2i)17-s + 0.999·18-s + (−1.98 − 1.14i)19-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.499 − 0.288i)3-s + (−0.249 + 0.433i)4-s + (0.353 + 0.204i)6-s + (0.249 − 0.432i)7-s − 0.353·8-s + (0.166 − 0.288i)9-s + (−1.20 + 0.695i)11-s + 0.288i·12-s + (−0.460 + 0.887i)13-s + 0.353·14-s + (−0.125 − 0.216i)16-s + (−0.840 − 0.485i)17-s + 0.235·18-s + (−0.455 − 0.262i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0280i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0280i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $-0.999 - 0.0280i$
Analytic conductor: \(15.5708\)
Root analytic conductor: \(3.94598\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1950} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1950,\ (\ :1/2),\ -0.999 - 0.0280i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7671127658\)
\(L(\frac12)\) \(\approx\) \(0.7671127658\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 + (-0.866 + 0.5i)T \)
5 \( 1 \)
13 \( 1 + (1.66 - 3.20i)T \)
good7 \( 1 + (-0.661 + 1.14i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (3.99 - 2.30i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (3.46 + 2i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.98 + 1.14i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (7.50 - 4.33i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.01 + 1.75i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 10.1iT - 31T^{2} \)
37 \( 1 + (3.40 + 5.89i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.02 - 2.32i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-7.45 - 4.30i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 9.10T + 47T^{2} \)
53 \( 1 - 0.826iT - 53T^{2} \)
59 \( 1 + (2.72 + 1.57i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.267 - 0.464i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.59 - 2.75i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (9.81 + 5.66i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 6.28T + 73T^{2} \)
79 \( 1 + 2.96T + 79T^{2} \)
83 \( 1 - 15.8T + 83T^{2} \)
89 \( 1 + (10.2 - 5.93i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (4.40 - 7.63i)T + (-48.5 - 84.0i)T^{2} \)
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

−9.357017903191930576466610296628, −8.700701835040998533337133037303, −7.71109737198996257514220413584, −7.35812516908237133809820805671, −6.60174511895894751958948078545, −5.55964390771386580506730657572, −4.63655672777321322962277780621, −4.04311415011649021824513506879, −2.74415414947250954553693225577, −1.86772957298661476980149687598, 0.20419612660738331341979579276, 2.13368973954189414687213192633, 2.63295046531469178458437837893, 3.75414760538866595884003854034, 4.54803889187196892679997572793, 5.53524781463784484045264738107, 6.06351697307381441288216316886, 7.44384976126499620687956069474, 8.294842802932128390286787698926, 8.657207574470019531397702682163

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