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Properties

Label 2-1890-63.47-c1-0-17
Degree $2$
Conductor $1890$
Sign $0.859 + 0.511i$
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 + (0.104 − 2.64i)7-s + 0.999i·8-s + (0.866 − 0.5i)10-s + 6.11i·11-s + (3.86 − 2.23i)13-s + (1.23 + 2.34i)14-s + (−0.5 − 0.866i)16-s + (−1.11 − 1.93i)17-s + (−0.623 − 0.360i)19-s + (−0.499 + 0.866i)20-s + (−3.05 − 5.29i)22-s − 6.71i·23-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s − 0.447·5-s + (0.0396 − 0.999i)7-s + 0.353i·8-s + (0.273 − 0.158i)10-s + 1.84i·11-s + (1.07 − 0.619i)13-s + (0.328 + 0.625i)14-s + (−0.125 − 0.216i)16-s + (−0.270 − 0.469i)17-s + (−0.143 − 0.0826i)19-s + (−0.111 + 0.193i)20-s + (−0.651 − 1.12i)22-s − 1.40i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 + 0.511i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.859 + 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1890\)    =    \(2 \cdot 3^{3} \cdot 5 \cdot 7\)
Sign: $0.859 + 0.511i$
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: \((2,\ 1890,\ (\ :1/2),\ 0.859 + 0.511i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.096975934\)
\(L(\frac12)\) \(\approx\) \(1.096975934\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.104 + 2.64i)T \)
good11 \( 1 - 6.11iT - 11T^{2} \)
13 \( 1 + (-3.86 + 2.23i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.11 + 1.93i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.623 + 0.360i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 6.71iT - 23T^{2} \)
29 \( 1 + (0.633 + 0.365i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-5.28 - 3.05i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.53 - 7.85i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.713 + 1.23i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.23 + 2.14i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.18 - 7.25i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.193 - 0.111i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.88 + 8.45i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.59 - 0.919i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.54 + 11.3i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 7.83iT - 71T^{2} \)
73 \( 1 + (-10.7 + 6.21i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (7.49 + 12.9i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.95 - 3.37i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-5.83 + 10.1i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-10.9 - 6.33i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.086560209797396502760592866025, −8.243965982396511946677931330118, −7.63729125647576820080743889517, −6.85959645971098815334393748208, −6.36538737040985310139679301303, −4.89459466853735111250428920212, −4.42267070123596186301987918351, −3.23693607567436149261587218153, −1.88299824590565421692289641726, −0.62887638385203418447665147332, 1.00112932557955534458743610550, 2.27139500526012043036040952259, 3.38500097108966859465752976481, 3.98050538527908588258187250616, 5.52883847198507458285476320537, 6.02850134638401408297873275017, 6.99370085879516020361782560096, 8.110749998617895660585271037994, 8.606165645714415095557916180456, 9.013171700893138837149455885214

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