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Properties

Label 2-1890-45.4-c1-0-23
Degree $2$
Conductor $1890$
Sign $0.849 + 0.527i$
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 + (1.91 + 1.15i)5-s + (0.866 − 0.5i)7-s − 0.999i·8-s + (2.23 + 0.0373i)10-s + (0.0898 + 0.155i)11-s + (1.02 + 0.589i)13-s + (0.499 − 0.866i)14-s + (−0.5 − 0.866i)16-s − 4.49i·17-s + 3.72·19-s + (1.95 − 1.08i)20-s + (0.155 + 0.0898i)22-s + (2.37 + 1.37i)23-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (0.857 + 0.514i)5-s + (0.327 − 0.188i)7-s − 0.353i·8-s + (0.707 + 0.0118i)10-s + (0.0271 + 0.0469i)11-s + (0.283 + 0.163i)13-s + (0.133 − 0.231i)14-s + (−0.125 − 0.216i)16-s − 1.09i·17-s + 0.854·19-s + (0.437 − 0.242i)20-s + (0.0331 + 0.0191i)22-s + (0.495 + 0.285i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.849 + 0.527i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.849 + 0.527i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1890\)    =    \(2 \cdot 3^{3} \cdot 5 \cdot 7\)
Sign: $0.849 + 0.527i$
Analytic conductor: \(15.0917\)
Root analytic conductor: \(3.88480\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1890} (1009, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1890,\ (\ :1/2),\ 0.849 + 0.527i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.202732290\)
\(L(\frac12)\) \(\approx\) \(3.202732290\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.91 - 1.15i)T \)
7 \( 1 + (-0.866 + 0.5i)T \)
good11 \( 1 + (-0.0898 - 0.155i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.02 - 0.589i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 4.49iT - 17T^{2} \)
19 \( 1 - 3.72T + 19T^{2} \)
23 \( 1 + (-2.37 - 1.37i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.52 + 2.63i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.23 - 3.86i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 1.15iT - 37T^{2} \)
41 \( 1 + (-4.91 + 8.51i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.839 + 0.484i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.61 - 0.934i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 0.659iT - 53T^{2} \)
59 \( 1 + (-3.56 + 6.17i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.48 - 7.77i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.92 - 2.84i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 12.6T + 71T^{2} \)
73 \( 1 - 10.4iT - 73T^{2} \)
79 \( 1 + (0.505 + 0.875i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (14.1 - 8.16i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 11.9T + 89T^{2} \)
97 \( 1 + (7.48 - 4.31i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.497095139344566605211696067658, −8.446701299660534901095394200400, −7.19847136729360155897621520770, −6.88172401754212766851609378840, −5.63989901994946407277047372505, −5.28503814495582608536121682284, −4.14520169936887430313132378165, −3.14506803615335219021531881871, −2.30257457829608502236724233676, −1.16364672815490316907629070475, 1.29392738963229965038342586601, 2.38659083224481914361000871787, 3.52497656124431042377856864666, 4.53978581833798831770912896999, 5.35077879305095730403921381632, 5.94191909012767125852626251326, 6.71013321420412965805304489320, 7.75607381939230281994255433609, 8.438017383627225276686639924273, 9.210369077193305525370593157384

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