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Properties

Label 2-1890-315.79-c1-0-24
Degree $2$
Conductor $1890$
Sign $0.711 + 0.702i$
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 + (−1.82 + 1.28i)5-s + (2.61 − 0.378i)7-s − 0.999i·8-s + (−0.937 + 2.03i)10-s − 1.86·11-s + (−2.13 + 1.23i)13-s + (2.07 − 1.63i)14-s + (−0.5 − 0.866i)16-s + (6.17 − 3.56i)17-s + (2.57 − 4.45i)19-s + (0.203 + 2.22i)20-s + (−1.61 + 0.930i)22-s + 4.03i·23-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (−0.816 + 0.576i)5-s + (0.989 − 0.143i)7-s − 0.353i·8-s + (−0.296 + 0.642i)10-s − 0.560·11-s + (−0.592 + 0.342i)13-s + (0.555 − 0.437i)14-s + (−0.125 − 0.216i)16-s + (1.49 − 0.864i)17-s + (0.590 − 1.02i)19-s + (0.0455 + 0.497i)20-s + (−0.343 + 0.198i)22-s + 0.840i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.711 + 0.702i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.711 + 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1890\)    =    \(2 \cdot 3^{3} \cdot 5 \cdot 7\)
Sign: $0.711 + 0.702i$
Analytic conductor: \(15.0917\)
Root analytic conductor: \(3.88480\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1890} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1890,\ (\ :1/2),\ 0.711 + 0.702i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.378629818\)
\(L(\frac12)\) \(\approx\) \(2.378629818\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.82 - 1.28i)T \)
7 \( 1 + (-2.61 + 0.378i)T \)
good11 \( 1 + 1.86T + 11T^{2} \)
13 \( 1 + (2.13 - 1.23i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-6.17 + 3.56i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.57 + 4.45i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 4.03iT - 23T^{2} \)
29 \( 1 + (-0.563 + 0.976i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.0124 - 0.0215i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.142 + 0.0823i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.73 - 4.73i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.90 - 3.40i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-10.8 + 6.26i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.19 + 1.26i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.32 + 7.48i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.21 - 5.57i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (13.6 + 7.87i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 10.0T + 71T^{2} \)
73 \( 1 + (-8.79 + 5.07i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.33 - 10.9i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (14.2 + 8.24i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (2.70 - 4.68i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.56 + 2.05i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.273660803001964353396091219538, −8.058915967215650989499131533947, −7.49024939856593716365006680130, −6.96501003688756565408120445789, −5.60776219376956786927432787140, −4.99640613192614593543500968514, −4.17648702492194704360195884121, −3.17481502764737024596550975654, −2.39183988994232669319921721070, −0.877392747465490270393065421371, 1.13509977992177248375620538211, 2.55562204314693669413291685533, 3.72355751522746617271050149684, 4.40896541637972502293860210801, 5.41667966039199092596722333246, 5.69079415384990613703966819219, 7.19600060268715774902278504480, 7.82643848954962147606240561767, 8.170461480581089285578160079099, 9.071372642344467848082100134170

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