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Properties

Label 2-1890-63.47-c1-0-5
Degree $2$
Conductor $1890$
Sign $0.506 - 0.862i$
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 + (1.16 + 2.37i)7-s − 0.999i·8-s + (−0.866 + 0.5i)10-s − 1.71i·11-s + (−4.83 + 2.78i)13-s + (2.19 + 1.47i)14-s + (−0.5 − 0.866i)16-s + (2.40 + 4.17i)17-s + (1.89 + 1.09i)19-s + (−0.499 + 0.866i)20-s + (−0.858 − 1.48i)22-s + 4.96i·23-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s − 0.447·5-s + (0.438 + 0.898i)7-s − 0.353i·8-s + (−0.273 + 0.158i)10-s − 0.517i·11-s + (−1.33 + 0.773i)13-s + (0.586 + 0.395i)14-s + (−0.125 − 0.216i)16-s + (0.584 + 1.01i)17-s + (0.434 + 0.250i)19-s + (−0.111 + 0.193i)20-s + (−0.183 − 0.317i)22-s + 1.03i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.506 - 0.862i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.506 - 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1890\)    =    \(2 \cdot 3^{3} \cdot 5 \cdot 7\)
Sign: $0.506 - 0.862i$
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: \((2,\ 1890,\ (\ :1/2),\ 0.506 - 0.862i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.879288900\)
\(L(\frac12)\) \(\approx\) \(1.879288900\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.16 - 2.37i)T \)
good11 \( 1 + 1.71iT - 11T^{2} \)
13 \( 1 + (4.83 - 2.78i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.40 - 4.17i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.89 - 1.09i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 4.96iT - 23T^{2} \)
29 \( 1 + (7.26 + 4.19i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-5.65 - 3.26i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.51 - 7.81i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.42 - 5.92i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.01 + 6.95i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.64 - 4.58i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.80 - 2.77i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.94 + 8.56i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-11.9 + 6.92i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.35 - 11.0i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 9.04iT - 71T^{2} \)
73 \( 1 + (2.87 - 1.65i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.63 + 2.82i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.50 - 9.53i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-0.463 + 0.803i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-10.6 - 6.13i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.518333688938891967422735504193, −8.506264731677994459928637799428, −7.81487938548885230230090247074, −6.94420517451766124441266364854, −5.89758273938557432409658976516, −5.30127832441440569655359087105, −4.41441704488380812853840631937, −3.49460854201652866427736004201, −2.52142437847962470113298865922, −1.48159682054378343087060994174, 0.55690815346334921264118058431, 2.29994220733079599644228547936, 3.30033848755849739137172416998, 4.32038104146416992742796434006, 4.93166023470410049195747446267, 5.68865445459909068300900433540, 7.08124902675442676693573777831, 7.33079261812165848048098578070, 7.934322138531798095755190797086, 9.041854510872382438482663242674

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