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Properties

Label 2-1890-63.59-c1-0-6
Degree $2$
Conductor $1890$
Sign $0.887 - 0.461i$
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 + (−2.46 + 0.968i)7-s − 0.999i·8-s + (0.866 + 0.5i)10-s + 0.656i·11-s + (2.39 + 1.38i)13-s + (2.61 + 0.392i)14-s + (−0.5 + 0.866i)16-s + (3.36 − 5.83i)17-s + (−0.633 + 0.365i)19-s + (−0.499 − 0.866i)20-s + (0.328 − 0.568i)22-s − 7.20i·23-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s − 0.447·5-s + (−0.930 + 0.366i)7-s − 0.353i·8-s + (0.273 + 0.158i)10-s + 0.197i·11-s + (0.664 + 0.383i)13-s + (0.699 + 0.104i)14-s + (−0.125 + 0.216i)16-s + (0.817 − 1.41i)17-s + (−0.145 + 0.0839i)19-s + (−0.111 − 0.193i)20-s + (0.0699 − 0.121i)22-s − 1.50i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.887 - 0.461i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.887 - 0.461i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1890\)    =    \(2 \cdot 3^{3} \cdot 5 \cdot 7\)
Sign: $0.887 - 0.461i$
Analytic conductor: \(15.0917\)
Root analytic conductor: \(3.88480\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1890} (1151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1890,\ (\ :1/2),\ 0.887 - 0.461i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8779721224\)
\(L(\frac12)\) \(\approx\) \(0.8779721224\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.46 - 0.968i)T \)
good11 \( 1 - 0.656iT - 11T^{2} \)
13 \( 1 + (-2.39 - 1.38i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.36 + 5.83i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.633 - 0.365i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 7.20iT - 23T^{2} \)
29 \( 1 + (6.21 - 3.58i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (6.93 - 4.00i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.05 - 1.81i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.35 + 5.81i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-6.29 - 10.8i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.92 - 5.07i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.77 - 3.33i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.62 - 8.01i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.35 - 0.782i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.28 - 5.68i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 1.54iT - 71T^{2} \)
73 \( 1 + (-5.95 - 3.43i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.53 + 13.0i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-8.71 - 15.0i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.32 - 2.29i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-4.02 + 2.32i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.186720379372852135511910112581, −8.787242551794550889738505916832, −7.70330140320666957448685036054, −7.07584007249862382240516648399, −6.27262112291271384871661960124, −5.27631920081380900578529662415, −4.09876117422557274326823829560, −3.24521623483256853786274613467, −2.38478088953340042364976681586, −0.873919036146189984327102418539, 0.54357708668208198807418365953, 1.92525508616410860208644250640, 3.54086535701103374365785506185, 3.83041406994433050490332896103, 5.52949475331457863718512062386, 5.94867319489557349323194911569, 6.94128198926484381703344146640, 7.68695841208882222124402289236, 8.244620777383340345220562577472, 9.238278891418160661964472525105

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