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Properties

Label 2-1890-63.47-c1-0-15
Degree $2$
Conductor $1890$
Sign $0.832 + 0.554i$
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 + 0.211i)7-s − 0.999i·8-s + (−0.866 + 0.5i)10-s + 4.91i·11-s + (3.91 − 2.26i)13-s + (−2.17 + 1.50i)14-s + (−0.5 − 0.866i)16-s + (−2.06 − 3.57i)17-s + (6.89 + 3.98i)19-s + (−0.499 + 0.866i)20-s + (2.45 + 4.25i)22-s − 5.03i·23-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s − 0.447·5-s + (−0.996 + 0.0800i)7-s − 0.353i·8-s + (−0.273 + 0.158i)10-s + 1.48i·11-s + (1.08 − 0.627i)13-s + (−0.582 + 0.401i)14-s + (−0.125 − 0.216i)16-s + (−0.501 − 0.868i)17-s + (1.58 + 0.913i)19-s + (−0.111 + 0.193i)20-s + (0.523 + 0.907i)22-s − 1.05i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.832 + 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1890\)    =    \(2 \cdot 3^{3} \cdot 5 \cdot 7\)
Sign: $0.832 + 0.554i$
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: \((2,\ 1890,\ (\ :1/2),\ 0.832 + 0.554i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.157462968\)
\(L(\frac12)\) \(\approx\) \(2.157462968\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.63 - 0.211i)T \)
good11 \( 1 - 4.91iT - 11T^{2} \)
13 \( 1 + (-3.91 + 2.26i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.06 + 3.57i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-6.89 - 3.98i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 5.03iT - 23T^{2} \)
29 \( 1 + (-2.46 - 1.42i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.36 - 0.789i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.05 + 8.76i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.31 - 7.47i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.88 - 4.99i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.227 + 0.393i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-10.7 + 6.21i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.48 + 9.49i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.66 + 2.11i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.96 - 3.40i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 8.14iT - 71T^{2} \)
73 \( 1 + (-2.06 + 1.19i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.86 + 8.42i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.42 + 2.46i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (2.95 - 5.11i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.74 - 1.00i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.431486361385343422225716351718, −8.339818194982905995044642437585, −7.39590236294968161374229263839, −6.74062842157565285521337329753, −5.89274365810428759985976329811, −4.98185660977034505576152021629, −4.09486931357413285956308994480, −3.28365173477667107692239694678, −2.40851530955704720815311559222, −0.893441890432408069040543537241, 0.958466680001847334812639710463, 2.79943990005213615587814321403, 3.54325970492561461713354924650, 4.15336784287072517334517406370, 5.46679542097746214812708108582, 6.07244678085846812099127373686, 6.78469757658846257305771263961, 7.57910423991165122766859759652, 8.591588442319190793427449819632, 9.006463629845426313065146857367

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