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Properties

Label 2-1890-63.47-c1-0-18
Degree $2$
Conductor $1890$
Sign $0.818 + 0.574i$
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 + (2.26 − 1.36i)7-s + 0.999i·8-s + (0.866 − 0.5i)10-s − 3.46i·11-s + (−0.584 + 0.337i)13-s + (−1.28 + 2.31i)14-s + (−0.5 − 0.866i)16-s + (2.09 + 3.62i)17-s + (1.18 + 0.683i)19-s + (−0.499 + 0.866i)20-s + (1.73 + 3.00i)22-s + 2.10i·23-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s − 0.447·5-s + (0.856 − 0.515i)7-s + 0.353i·8-s + (0.273 − 0.158i)10-s − 1.04i·11-s + (−0.162 + 0.0936i)13-s + (−0.342 + 0.618i)14-s + (−0.125 − 0.216i)16-s + (0.507 + 0.878i)17-s + (0.271 + 0.156i)19-s + (−0.111 + 0.193i)20-s + (0.369 + 0.640i)22-s + 0.437i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.818 + 0.574i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.818 + 0.574i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1890\)    =    \(2 \cdot 3^{3} \cdot 5 \cdot 7\)
Sign: $0.818 + 0.574i$
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: \((2,\ 1890,\ (\ :1/2),\ 0.818 + 0.574i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.233115820\)
\(L(\frac12)\) \(\approx\) \(1.233115820\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.26 + 1.36i)T \)
good11 \( 1 + 3.46iT - 11T^{2} \)
13 \( 1 + (0.584 - 0.337i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.09 - 3.62i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.18 - 0.683i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 2.10iT - 23T^{2} \)
29 \( 1 + (-4.77 - 2.75i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.25 + 0.724i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.59 + 7.95i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (4.79 + 8.30i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.38 - 2.39i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.50 + 4.33i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.77 + 3.90i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.15 + 3.73i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.15 - 2.97i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.90 - 3.29i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 15.0iT - 71T^{2} \)
73 \( 1 + (-2.54 + 1.46i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.65 + 11.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.69 + 2.93i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-3.11 + 5.39i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.750 + 0.433i)T + (48.5 + 84.0i)T^{2} \)
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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

−8.801964282204585125959142611551, −8.427282261472761632388106856256, −7.62811222920015680456658580177, −7.04898870881735362090935196283, −5.95608489268852507816852636992, −5.27116195703181278068715658238, −4.17582696782683440040580222310, −3.29043104587800133649775395235, −1.81679680000397006773888416761, −0.67659271913225080022766946291, 1.09271791836872057860886111775, 2.28100389431108132564900146613, 3.14800625056437957939945203958, 4.52462626758276265255681537283, 4.97190982122932045970373880538, 6.26777730463046019681190411495, 7.21577908729017754345418890728, 7.85802720130477922287941391456, 8.455505185661126819869749941895, 9.367752225346829071733213450851

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