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Properties

Label 2-1890-63.47-c1-0-10
Degree $2$
Conductor $1890$
Sign $0.934 - 0.356i$
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 + (2.60 + 0.461i)7-s − 0.999i·8-s + (−0.866 + 0.5i)10-s + 5.17i·11-s + (0.604 − 0.349i)13-s + (2.48 − 0.903i)14-s + (−0.5 − 0.866i)16-s + (1.77 + 3.07i)17-s + (−4.54 − 2.62i)19-s + (−0.499 + 0.866i)20-s + (2.58 + 4.48i)22-s + 5.75i·23-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s − 0.447·5-s + (0.984 + 0.174i)7-s − 0.353i·8-s + (−0.273 + 0.158i)10-s + 1.56i·11-s + (0.167 − 0.0968i)13-s + (0.664 − 0.241i)14-s + (−0.125 − 0.216i)16-s + (0.430 + 0.745i)17-s + (−1.04 − 0.601i)19-s + (−0.111 + 0.193i)20-s + (0.552 + 0.956i)22-s + 1.20i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.934 - 0.356i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.934 - 0.356i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1890\)    =    \(2 \cdot 3^{3} \cdot 5 \cdot 7\)
Sign: $0.934 - 0.356i$
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: \((2,\ 1890,\ (\ :1/2),\ 0.934 - 0.356i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.499571150\)
\(L(\frac12)\) \(\approx\) \(2.499571150\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.60 - 0.461i)T \)
good11 \( 1 - 5.17iT - 11T^{2} \)
13 \( 1 + (-0.604 + 0.349i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.77 - 3.07i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.54 + 2.62i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 5.75iT - 23T^{2} \)
29 \( 1 + (-4.60 - 2.65i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.81 - 2.77i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.06 + 8.77i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.18 - 5.52i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.94 - 10.3i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.08 + 7.06i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (8.82 - 5.09i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.679 + 1.17i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.28 + 3.05i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.21 + 3.82i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 4.76iT - 71T^{2} \)
73 \( 1 + (3.19 - 1.84i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.81 - 10.0i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.80 + 13.5i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (2.57 - 4.46i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-15.5 - 9.00i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.392288135500624326248140836277, −8.339411698947663157224223819528, −7.72891132326191867447249961593, −6.86428702654125011677953629464, −5.98885996312034556628604901928, −4.78689087070619161714996690913, −4.58893537859084471809308442979, −3.47195168865521299924892774384, −2.27833638318085872803326620336, −1.38699891445562905625442738860, 0.809736117804233504919854860752, 2.41060606598241750191516452550, 3.43053833960316533151570456170, 4.34489951291293064452932965033, 5.00879041612605445689529273837, 6.04246165533189437653336924593, 6.60689726724997667743598335939, 7.75776442299232071219980583988, 8.291033687608724714726539331961, 8.736020238156274294337509465157

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