Properties

Label 2-1890-63.59-c1-0-11
Degree $2$
Conductor $1890$
Sign $-0.102 - 0.994i$
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 + (−1.57 − 2.12i)7-s + 0.999i·8-s + (0.866 + 0.5i)10-s + 5.66i·11-s + (2.43 + 1.40i)13-s + (−0.300 − 2.62i)14-s + (−0.5 + 0.866i)16-s + (−3.51 + 6.08i)17-s + (3.76 − 2.17i)19-s + (0.499 + 0.866i)20-s + (−2.83 + 4.90i)22-s − 4.07i·23-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + 0.447·5-s + (−0.595 − 0.803i)7-s + 0.353i·8-s + (0.273 + 0.158i)10-s + 1.70i·11-s + (0.675 + 0.389i)13-s + (−0.0802 − 0.702i)14-s + (−0.125 + 0.216i)16-s + (−0.851 + 1.47i)17-s + (0.863 − 0.498i)19-s + (0.111 + 0.193i)20-s + (−0.603 + 1.04i)22-s − 0.849i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.102 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.102 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1890\)    =    \(2 \cdot 3^{3} \cdot 5 \cdot 7\)
Sign: $-0.102 - 0.994i$
Analytic conductor: \(15.0917\)
Root analytic conductor: \(3.88480\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1890} (1151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1890,\ (\ :1/2),\ -0.102 - 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.322274157\)
\(L(\frac12)\) \(\approx\) \(2.322274157\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.57 + 2.12i)T \)
good11 \( 1 - 5.66iT - 11T^{2} \)
13 \( 1 + (-2.43 - 1.40i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.51 - 6.08i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.76 + 2.17i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 4.07iT - 23T^{2} \)
29 \( 1 + (5.28 - 3.05i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.319 + 0.184i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.783 - 1.35i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.39 - 2.41i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.18 - 5.51i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.59 + 4.49i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.02 - 1.74i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.32 - 9.22i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-12.6 - 7.32i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.36 + 2.36i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 9.06iT - 71T^{2} \)
73 \( 1 + (14.3 + 8.29i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.37 + 7.57i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-8.34 - 14.4i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-0.332 - 0.575i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.68 + 0.971i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.423618347994250555008612679434, −8.659043305438314864341421264410, −7.57702439405385264907087741019, −6.89153197786466251618453473977, −6.40646103082331481440166165984, −5.41443210183072217703440700094, −4.35822333522649854326364619842, −3.92760114669765130139310258288, −2.62001519520502377897447503342, −1.52598879643793735976422669443, 0.69025509792455067382845696642, 2.20676795058511762187990028472, 3.13669010318901898556877950893, 3.73279524887458207463156260904, 5.23873837992026162794708816437, 5.66760240637918361700232999894, 6.31154204298722576617813385055, 7.30076934778005176806309554151, 8.406573712798218023565579772424, 9.162619637520546403420202407774

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