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Properties

Label 2-1890-63.59-c1-0-9
Degree $2$
Conductor $1890$
Sign $-0.551 - 0.834i$
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 + (1.49 + 2.18i)7-s + 0.999i·8-s + (−0.866 − 0.5i)10-s + 3.47i·11-s + (0.972 + 0.561i)13-s + (0.205 + 2.63i)14-s + (−0.5 + 0.866i)16-s + (0.795 − 1.37i)17-s + (0.478 − 0.276i)19-s + (−0.499 − 0.866i)20-s + (−1.73 + 3.01i)22-s + 0.00487i·23-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s − 0.447·5-s + (0.565 + 0.824i)7-s + 0.353i·8-s + (−0.273 − 0.158i)10-s + 1.04i·11-s + (0.269 + 0.155i)13-s + (0.0550 + 0.704i)14-s + (−0.125 + 0.216i)16-s + (0.192 − 0.334i)17-s + (0.109 − 0.0633i)19-s + (−0.111 − 0.193i)20-s + (−0.370 + 0.642i)22-s + 0.00101i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.551 - 0.834i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.551 - 0.834i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1890\)    =    \(2 \cdot 3^{3} \cdot 5 \cdot 7\)
Sign: $-0.551 - 0.834i$
Analytic conductor: \(15.0917\)
Root analytic conductor: \(3.88480\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1890} (1151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1890,\ (\ :1/2),\ -0.551 - 0.834i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.132783594\)
\(L(\frac12)\) \(\approx\) \(2.132783594\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.49 - 2.18i)T \)
good11 \( 1 - 3.47iT - 11T^{2} \)
13 \( 1 + (-0.972 - 0.561i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.795 + 1.37i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.478 + 0.276i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 0.00487iT - 23T^{2} \)
29 \( 1 + (2.13 - 1.22i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (5.51 - 3.18i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.212 - 0.368i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.76 + 3.05i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.300 + 0.520i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.22 - 9.05i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.21 - 1.28i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.70 + 2.94i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.05 + 2.34i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.77 - 11.7i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 14.9iT - 71T^{2} \)
73 \( 1 + (4.45 + 2.57i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.37 + 4.11i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.21 + 5.57i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-3.59 - 6.23i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-14.3 + 8.26i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.276837623372591515478675492556, −8.659862810738610999638997765258, −7.71662922128813548693662993146, −7.21281956843566504011581778304, −6.24609781779946910690320946703, −5.33090210620634620078481089822, −4.72188241714245387273755386281, −3.81392612352224185467791501977, −2.73414622892777417213383293512, −1.67320345934151650582378081705, 0.63322791048566442654882726167, 1.86134102941120442484993750141, 3.29421136061088330119299067854, 3.82470452301239876058609017846, 4.75844883159420976258478076507, 5.63279964804009100382992106252, 6.44637000753088627662475957168, 7.44669667586469289271952308911, 8.060071351836250839067906066764, 8.904381155379190247074428831522

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