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Properties

Label 2-1890-315.209-c1-0-43
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

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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: $\chi_{1890} (629, \cdot )$
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 \( 1 + (4.73 - 8.19i)T + (-48.5 - 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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