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Properties

Label 2-1890-63.59-c1-0-18
Degree $2$
Conductor $1890$
Sign $0.306 + 0.951i$
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 + (1.13 − 2.38i)7-s − 0.999i·8-s + (0.866 + 0.5i)10-s − 1.06i·11-s + (3.43 + 1.98i)13-s + (−2.18 + 1.49i)14-s + (−0.5 + 0.866i)16-s + (−3.26 + 5.65i)17-s + (1.00 − 0.582i)19-s + (−0.499 − 0.866i)20-s + (−0.533 + 0.924i)22-s − 7.37i·23-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s − 0.447·5-s + (0.430 − 0.902i)7-s − 0.353i·8-s + (0.273 + 0.158i)10-s − 0.321i·11-s + (0.952 + 0.549i)13-s + (−0.582 + 0.400i)14-s + (−0.125 + 0.216i)16-s + (−0.792 + 1.37i)17-s + (0.231 − 0.133i)19-s + (−0.111 − 0.193i)20-s + (−0.113 + 0.197i)22-s − 1.53i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.306 + 0.951i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.306 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1890\)    =    \(2 \cdot 3^{3} \cdot 5 \cdot 7\)
Sign: $0.306 + 0.951i$
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: \((2,\ 1890,\ (\ :1/2),\ 0.306 + 0.951i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.190031800\)
\(L(\frac12)\) \(\approx\) \(1.190031800\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.13 + 2.38i)T \)
good11 \( 1 + 1.06iT - 11T^{2} \)
13 \( 1 + (-3.43 - 1.98i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.26 - 5.65i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.00 + 0.582i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 7.37iT - 23T^{2} \)
29 \( 1 + (-5.89 + 3.40i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.61 + 2.08i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.89 - 5.02i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (5.22 - 9.05i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.665 - 1.15i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-6.02 + 10.4i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (7.55 + 4.36i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.63 - 8.02i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.02 - 1.74i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.98 + 5.16i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 0.707iT - 71T^{2} \)
73 \( 1 + (6.25 + 3.60i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.43 + 5.95i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.68 + 4.65i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-2.09 - 3.62i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-4.59 + 2.65i)T + (48.5 - 84.0i)T^{2} \)
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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

−8.749136847806863547689145003026, −8.451859983204078947127289268381, −7.75556770497063039461943995756, −6.64206869623071465015833914777, −6.26897622320476035433691872793, −4.58849937505142646714537097285, −4.13256933350169092547879179006, −3.08556323179230550550456404280, −1.78606084574216938394689654256, −0.66987723691252796413711917531, 1.05464436951457359264256654687, 2.35711586306070503514036938827, 3.38805866598931183285116914702, 4.70045376812564385049304670480, 5.41651525337255339793167050409, 6.25893168736582654981232985168, 7.22029499239584839098772653095, 7.81598536649341724306871745509, 8.712926890794869620078867784968, 9.100710577094943354886875468202

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