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Properties

Label 2-1890-63.47-c1-0-7
Degree $2$
Conductor $1890$
Sign $0.242 - 0.970i$
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  + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s − 5-s + (2.63 − 0.194i)7-s + 0.999i·8-s + (0.866 − 0.5i)10-s + 2.09i·11-s + (−0.413 + 0.238i)13-s + (−2.18 + 1.48i)14-s + (−0.5 − 0.866i)16-s + (1.44 + 2.49i)17-s + (4.03 + 2.32i)19-s + (−0.499 + 0.866i)20-s + (−1.04 − 1.81i)22-s − 2.02i·23-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s − 0.447·5-s + (0.997 − 0.0735i)7-s + 0.353i·8-s + (0.273 − 0.158i)10-s + 0.630i·11-s + (−0.114 + 0.0662i)13-s + (−0.584 + 0.397i)14-s + (−0.125 − 0.216i)16-s + (0.349 + 0.605i)17-s + (0.925 + 0.534i)19-s + (−0.111 + 0.193i)20-s + (−0.222 − 0.386i)22-s − 0.422i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.242 - 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.242 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1890\)    =    \(2 \cdot 3^{3} \cdot 5 \cdot 7\)
Sign: $0.242 - 0.970i$
Analytic conductor: \(15.0917\)
Root analytic conductor: \(3.88480\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1890} (1601, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1890,\ (\ :1/2),\ 0.242 - 0.970i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.218700155\)
\(L(\frac12)\) \(\approx\) \(1.218700155\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 + T \)
7 \( 1 + (-2.63 + 0.194i)T \)
good11 \( 1 - 2.09iT - 11T^{2} \)
13 \( 1 + (0.413 - 0.238i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.44 - 2.49i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.03 - 2.32i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 2.02iT - 23T^{2} \)
29 \( 1 + (2.74 + 1.58i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.59 + 2.65i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.93 - 3.34i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.03 - 6.99i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.96 - 5.14i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.0525 + 0.0909i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.27 + 2.46i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.40 - 2.44i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.71 + 2.14i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.73 - 4.73i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 10.8iT - 71T^{2} \)
73 \( 1 + (-5.64 + 3.26i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-7.81 - 13.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.58 - 9.68i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-6.59 + 11.4i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-7.98 - 4.60i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.368202472958601879080167969588, −8.410455412501995493986649990056, −7.82389544197059318250465780511, −7.33202422501006673862727610895, −6.32060189533790167305603995334, −5.37481688743050583400721433980, −4.60355511766167552854003870804, −3.60312316955089753245020987964, −2.18819740759325137143142286290, −1.15816344960184713730409626760, 0.63951404566050460023332883429, 1.84946645319781072164880421134, 3.03737454698824049766874828358, 3.89119839734995657026202364424, 5.05133661413634635572286238894, 5.69674056068109813111814062759, 7.20425341781480437549494280725, 7.42013744345357520849367406980, 8.422332097006334914448476875191, 8.959719064566847943191586792340

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