↑

Properties

Label 2-1890-315.104-c1-0-0
Degree $2$
Conductor $1890$
Sign $-0.978 + 0.205i$
Analytic cond. $15.0917$
Root an. cond. $3.88480$
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 + (−0.499 + 0.866i)4-s + (−1.00 + 1.99i)5-s + (−2.45 + 0.977i)7-s + 0.999·8-s + (2.23 − 0.124i)10-s + (1.44 − 0.832i)11-s + (−1.04 + 1.81i)13-s + (2.07 + 1.64i)14-s + (−0.5 − 0.866i)16-s + 7.45i·17-s − 4.66i·19-s + (−1.22 − 1.87i)20-s + (−1.44 − 0.832i)22-s + (−2.82 + 4.89i)23-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.450 + 0.892i)5-s + (−0.929 + 0.369i)7-s + 0.353·8-s + (0.706 − 0.0394i)10-s + (0.434 − 0.250i)11-s + (−0.290 + 0.503i)13-s + (0.554 + 0.438i)14-s + (−0.125 − 0.216i)16-s + 1.80i·17-s − 1.06i·19-s + (−0.273 − 0.418i)20-s + (−0.307 − 0.177i)22-s + (−0.589 + 1.02i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 + 0.205i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.978 + 0.205i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1890\)    =    \(2 \cdot 3^{3} \cdot 5 \cdot 7\)
Sign: $-0.978 + 0.205i$
Analytic conductor: \(15.0917\)
Root analytic conductor: \(3.88480\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1890} (1259, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1890,\ (\ :1/2),\ -0.978 + 0.205i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.03341208965\)
\(L(\frac12)\) \(\approx\) \(0.03341208965\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 + (1.00 - 1.99i)T \)
7 \( 1 + (2.45 - 0.977i)T \)
good11 \( 1 + (-1.44 + 0.832i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.04 - 1.81i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 7.45iT - 17T^{2} \)
19 \( 1 + 4.66iT - 19T^{2} \)
23 \( 1 + (2.82 - 4.89i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.10 + 2.37i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.73 + 1.00i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 4.86iT - 37T^{2} \)
41 \( 1 + (4.07 - 7.05i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.25 + 1.30i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-8.90 + 5.13i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 13.8T + 53T^{2} \)
59 \( 1 + (0.0862 - 0.149i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.17 - 2.98i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.70 + 2.71i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 12.0iT - 71T^{2} \)
73 \( 1 + 4.79T + 73T^{2} \)
79 \( 1 + (6.30 + 10.9i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.24 - 1.29i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 11.2T + 89T^{2} \)
97 \( 1 + (4.73 + 8.19i)T + (-48.5 + 84.0i)T^{2} \)
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

−9.635118612544263080211039506485, −9.030679989860738849327796522449, −8.160416371817224918927736271176, −7.33236139850798102984146206113, −6.48523524907518261704066903011, −5.89837878928248076611210865818, −4.38194568567911290010119746134, −3.62889375678115607349769421969, −2.87423743100927294102292986542, −1.81317640831498333132932229778, 0.01534761420195545058061682205, 1.14689845701665713956859070709, 2.83338715965640280988296697840, 3.98055945122554889157569294306, 4.77683316406280999786190007180, 5.62310058638912345853190316671, 6.56304595292618315375191458007, 7.28470375844279715344845959363, 7.976975023319683112630117553814, 8.804662383266370116260500477362

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