Properties

Label 2-1890-315.104-c1-0-21
Degree $2$
Conductor $1890$
Sign $0.928 - 0.371i$
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  + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (2.16 + 0.548i)5-s + (−2.64 − 0.0312i)7-s + 0.999·8-s + (−0.609 − 2.15i)10-s + (2.30 − 1.33i)11-s + (1.15 − 2.00i)13-s + (1.29 + 2.30i)14-s + (−0.5 − 0.866i)16-s + 7.23i·17-s + 5.58i·19-s + (−1.55 + 1.60i)20-s + (−2.30 − 1.33i)22-s + (−0.354 + 0.613i)23-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.969 + 0.245i)5-s + (−0.999 − 0.0118i)7-s + 0.353·8-s + (−0.192 − 0.680i)10-s + (0.695 − 0.401i)11-s + (0.321 − 0.557i)13-s + (0.346 + 0.616i)14-s + (−0.125 − 0.216i)16-s + 1.75i·17-s + 1.28i·19-s + (−0.348 + 0.358i)20-s + (−0.491 − 0.283i)22-s + (−0.0738 + 0.127i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.928 - 0.371i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.928 - 0.371i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1890\)    =    \(2 \cdot 3^{3} \cdot 5 \cdot 7\)
Sign: $0.928 - 0.371i$
Analytic conductor: \(15.0917\)
Root analytic conductor: \(3.88480\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1890} (1259, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1890,\ (\ :1/2),\ 0.928 - 0.371i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.437975216\)
\(L(\frac12)\) \(\approx\) \(1.437975216\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 \)
5 \( 1 + (-2.16 - 0.548i)T \)
7 \( 1 + (2.64 + 0.0312i)T \)
good11 \( 1 + (-2.30 + 1.33i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.15 + 2.00i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 7.23iT - 17T^{2} \)
19 \( 1 - 5.58iT - 19T^{2} \)
23 \( 1 + (0.354 - 0.613i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.68 + 1.55i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (5.32 + 3.07i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 8.09iT - 37T^{2} \)
41 \( 1 + (-3.67 + 6.37i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (11.3 - 6.54i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.91 + 2.83i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 3.40T + 53T^{2} \)
59 \( 1 + (-1.98 + 3.43i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-8.66 + 5.00i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.96 - 2.86i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 9.75iT - 71T^{2} \)
73 \( 1 + 4.33T + 73T^{2} \)
79 \( 1 + (-7.29 - 12.6i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.25 + 1.30i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 5.66T + 89T^{2} \)
97 \( 1 + (-6.43 - 11.1i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.454689684259313640729020188859, −8.610051542690693894183943615406, −7.975676004715752231030210299555, −6.66680925197165216974263586427, −6.17807313828582882902616749134, −5.42711328905969245018669178473, −3.85962366561779776268004783454, −3.41441403535293775350884166835, −2.20382654914104043673810352514, −1.20512330250815787667933139816, 0.65990813826904820844490041674, 2.06273697027743099931781830760, 3.16922933205961279014309471996, 4.49676738118958909097240893718, 5.24837942052552409602539235525, 6.16356863303241930510513238598, 6.88345309589714837004900713766, 7.24127524774986944166853148143, 8.756133890468071775739573595086, 9.181137492579341318221256886119

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