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Properties

Label 2-1890-45.4-c1-0-27
Degree $2$
Conductor $1890$
Sign $-0.373 + 0.927i$
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 + (−2.18 + 0.482i)5-s + (0.866 − 0.5i)7-s − 0.999i·8-s + (−1.64 + 1.50i)10-s + (0.133 + 0.231i)11-s + (1.09 + 0.633i)13-s + (0.499 − 0.866i)14-s + (−0.5 − 0.866i)16-s − 4.03i·17-s + 1.58·19-s + (−0.674 + 2.13i)20-s + (0.231 + 0.133i)22-s + (−4.19 − 2.42i)23-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (−0.976 + 0.215i)5-s + (0.327 − 0.188i)7-s − 0.353i·8-s + (−0.521 + 0.477i)10-s + (0.0402 + 0.0696i)11-s + (0.304 + 0.175i)13-s + (0.133 − 0.231i)14-s + (−0.125 − 0.216i)16-s − 0.979i·17-s + 0.363·19-s + (−0.150 + 0.476i)20-s + (0.0492 + 0.0284i)22-s + (−0.874 − 0.504i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.373 + 0.927i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.373 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1890\)    =    \(2 \cdot 3^{3} \cdot 5 \cdot 7\)
Sign: $-0.373 + 0.927i$
Analytic conductor: \(15.0917\)
Root analytic conductor: \(3.88480\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1890} (1009, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1890,\ (\ :1/2),\ -0.373 + 0.927i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.727267503\)
\(L(\frac12)\) \(\approx\) \(1.727267503\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.18 - 0.482i)T \)
7 \( 1 + (-0.866 + 0.5i)T \)
good11 \( 1 + (-0.133 - 0.231i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.09 - 0.633i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 4.03iT - 17T^{2} \)
19 \( 1 - 1.58T + 19T^{2} \)
23 \( 1 + (4.19 + 2.42i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.45 + 4.25i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.169 + 0.293i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 2.43iT - 37T^{2} \)
41 \( 1 + (1.76 - 3.04i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.75 + 2.16i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-7.08 + 4.08i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 13.1iT - 53T^{2} \)
59 \( 1 + (-0.637 + 1.10i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.32 + 7.49i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (8.90 + 5.14i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 0.947T + 71T^{2} \)
73 \( 1 - 12.6iT - 73T^{2} \)
79 \( 1 + (6.28 + 10.8i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.16 + 4.13i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 10.8T + 89T^{2} \)
97 \( 1 + (-10.7 + 6.23i)T + (48.5 - 84.0i)T^{2} \)
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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

−8.983755107716363038830556032282, −8.061227419790501444908663159957, −7.38343719063929732161789775454, −6.62303249110026349598773840357, −5.63105382555656081402148426631, −4.67131110498583943967457623196, −4.01478426342118945073314881791, −3.16603249739970951435216406001, −2.06674437553605775378141523713, −0.53221304079571079588746834561, 1.41841591354535267614664601054, 2.89625031344434135183737783770, 3.84197727046032177314818193326, 4.44122362230495837689689875748, 5.48418771078395510524234501017, 6.12236661584908580500865199379, 7.25449647926608244976775443233, 7.76425097051918322256746403198, 8.527154245938010668424814307442, 9.181767305708020623118478564280

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