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Properties

Label 2-1890-63.47-c1-0-2
Degree $2$
Conductor $1890$
Sign $-0.872 - 0.489i$
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 + (−1.89 − 1.84i)7-s + 0.999i·8-s + (0.866 − 0.5i)10-s − 0.645i·11-s + (−0.230 + 0.133i)13-s + (2.56 + 0.647i)14-s + (−0.5 − 0.866i)16-s + (0.525 + 0.910i)17-s + (0.938 + 0.541i)19-s + (−0.499 + 0.866i)20-s + (0.322 + 0.559i)22-s − 3.36i·23-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s − 0.447·5-s + (−0.717 − 0.696i)7-s + 0.353i·8-s + (0.273 − 0.158i)10-s − 0.194i·11-s + (−0.0639 + 0.0369i)13-s + (0.685 + 0.173i)14-s + (−0.125 − 0.216i)16-s + (0.127 + 0.220i)17-s + (0.215 + 0.124i)19-s + (−0.111 + 0.193i)20-s + (0.0688 + 0.119i)22-s − 0.702i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.872 - 0.489i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.872 - 0.489i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1890\)    =    \(2 \cdot 3^{3} \cdot 5 \cdot 7\)
Sign: $-0.872 - 0.489i$
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: \((2,\ 1890,\ (\ :1/2),\ -0.872 - 0.489i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2179853806\)
\(L(\frac12)\) \(\approx\) \(0.2179853806\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.89 + 1.84i)T \)
good11 \( 1 + 0.645iT - 11T^{2} \)
13 \( 1 + (0.230 - 0.133i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.525 - 0.910i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.938 - 0.541i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 3.36iT - 23T^{2} \)
29 \( 1 + (2.95 + 1.70i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.57 - 1.48i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.37 + 2.37i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.16 + 3.75i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.04 - 8.74i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.0268 - 0.0465i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (10.1 - 5.86i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.85 + 4.93i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.06 + 1.77i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.61 - 6.25i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 10.7iT - 71T^{2} \)
73 \( 1 + (13.9 - 8.03i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.16 - 3.74i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.83 + 3.16i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (6.12 - 10.6i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (14.8 + 8.58i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.606000526590381755998080950419, −8.670648846756488045123694230877, −7.987881192831809460492628042340, −7.23019889194890315894152714978, −6.56339703658050103598948277860, −5.77566162239649434840690673056, −4.63567301628351131025613689039, −3.72821736836721703980695446849, −2.71734540897124148311736053441, −1.18190172535967149966721784887, 0.10718581391983359256009374550, 1.70735521774729992891638052728, 2.89586323530420963665769921047, 3.56862828239754555422186378046, 4.76643178043606230174463037364, 5.75748578442321203780277245576, 6.68133386235767950759024377830, 7.42373134041767211647983434414, 8.229335497355208775403395508863, 8.986570506033142143665156759133

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