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Properties

Label 2-1890-63.47-c1-0-19
Degree $2$
Conductor $1890$
Sign $0.885 + 0.464i$
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 + (1.07 − 2.41i)7-s − 0.999i·8-s + (0.866 − 0.5i)10-s + 4.51i·11-s + (1.92 − 1.11i)13-s + (−0.279 − 2.63i)14-s + (−0.5 − 0.866i)16-s + (3.31 + 5.74i)17-s + (6.71 + 3.87i)19-s + (0.499 − 0.866i)20-s + (2.25 + 3.90i)22-s + 5.02i·23-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + 0.447·5-s + (0.405 − 0.914i)7-s − 0.353i·8-s + (0.273 − 0.158i)10-s + 1.36i·11-s + (0.534 − 0.308i)13-s + (−0.0747 − 0.703i)14-s + (−0.125 − 0.216i)16-s + (0.803 + 1.39i)17-s + (1.53 + 0.888i)19-s + (0.111 − 0.193i)20-s + (0.480 + 0.832i)22-s + 1.04i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.885 + 0.464i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.885 + 0.464i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1890\)    =    \(2 \cdot 3^{3} \cdot 5 \cdot 7\)
Sign: $0.885 + 0.464i$
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: \((2,\ 1890,\ (\ :1/2),\ 0.885 + 0.464i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.094309698\)
\(L(\frac12)\) \(\approx\) \(3.094309698\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.07 + 2.41i)T \)
good11 \( 1 - 4.51iT - 11T^{2} \)
13 \( 1 + (-1.92 + 1.11i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.31 - 5.74i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-6.71 - 3.87i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 5.02iT - 23T^{2} \)
29 \( 1 + (1.08 + 0.623i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.32 + 2.49i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.475 - 0.822i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (4.31 + 7.46i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.87 + 8.43i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.53 + 4.38i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.68 + 2.70i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.63 + 2.83i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-9.88 + 5.70i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.76 - 4.79i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 9.57iT - 71T^{2} \)
73 \( 1 + (-1.11 + 0.642i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.98 - 3.43i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.71 - 9.90i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (8.89 - 15.3i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (7.24 + 4.18i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.506846372572043325714102270890, −8.222469538601271999496542467344, −7.47940561017851424149987272529, −6.83789613507291176816301087597, −5.58335047698374663377683956682, −5.29289573292338457304577736464, −3.89430496403184115631524906635, −3.62846944340430662379813676026, −1.99216382141873315673812454309, −1.29001398164293235094183597112, 1.14721313245433792093502015266, 2.73373136417430796213672238744, 3.18169675916692225170484784657, 4.59901791484296489185510312837, 5.40788132473079991756285203151, 5.84577335694979858037813377450, 6.78933153993374757008296594589, 7.64115642316171506060191596913, 8.556960852084779713110925216223, 9.089728197630202425347144505300

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