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Properties

Label 2-1890-63.47-c1-0-9
Degree $2$
Conductor $1890$
Sign $0.890 - 0.454i$
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 + (−2.11 + 1.59i)7-s − 0.999i·8-s + (0.866 − 0.5i)10-s − 0.767i·11-s + (−3.78 + 2.18i)13-s + (−1.03 + 2.43i)14-s + (−0.5 − 0.866i)16-s + (1.15 + 1.99i)17-s + (4.19 + 2.42i)19-s + (0.499 − 0.866i)20-s + (−0.383 − 0.664i)22-s + 5.59i·23-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + 0.447·5-s + (−0.798 + 0.602i)7-s − 0.353i·8-s + (0.273 − 0.158i)10-s − 0.231i·11-s + (−1.04 + 0.605i)13-s + (−0.276 + 0.650i)14-s + (−0.125 − 0.216i)16-s + (0.279 + 0.483i)17-s + (0.961 + 0.555i)19-s + (0.111 − 0.193i)20-s + (−0.0818 − 0.141i)22-s + 1.16i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.890 - 0.454i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.890 - 0.454i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1890\)    =    \(2 \cdot 3^{3} \cdot 5 \cdot 7\)
Sign: $0.890 - 0.454i$
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: \((2,\ 1890,\ (\ :1/2),\ 0.890 - 0.454i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.307999330\)
\(L(\frac12)\) \(\approx\) \(2.307999330\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.11 - 1.59i)T \)
good11 \( 1 + 0.767iT - 11T^{2} \)
13 \( 1 + (3.78 - 2.18i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.15 - 1.99i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.19 - 2.42i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 5.59iT - 23T^{2} \)
29 \( 1 + (-9.12 - 5.26i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-6.97 - 4.02i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.15 + 7.19i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.65 - 4.59i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.94 + 10.3i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.23 + 5.59i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (6.48 - 3.74i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.71 - 9.90i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (8.22 - 4.74i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.85 - 11.8i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 7.50iT - 71T^{2} \)
73 \( 1 + (-10.1 + 5.84i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.555 - 0.961i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.254 + 0.440i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (2.80 - 4.85i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (7.61 + 4.39i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.449596546504955237409920871633, −8.710918015610209325277397962043, −7.53486192315380346305521819778, −6.75142529070858948989010794575, −5.90451739488399616235801286529, −5.33124887876881087239521660401, −4.34159368926881334108994060699, −3.22144862048944276023635312004, −2.57517378147374015554593766617, −1.32170536145308806552565877001, 0.73091256008237115472672154669, 2.63768737873729389724314936003, 3.06563036042800264751473258874, 4.56881178803987210337068344187, 4.85397720944145887303009792984, 6.26417727901243752651301023772, 6.45902486191010470431765188492, 7.61359079397626337866828577809, 8.016222102148728214538592965001, 9.455310393390703124407756013978

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