↑

Properties

Label 2-1710-19.7-c1-0-28
Degree $2$
Conductor $1710$
Sign $-0.805 + 0.592i$
Analytic cond. $13.6544$
Root an. cond. $3.69518$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (0.5 + 0.866i)5-s − 7-s + 0.999·8-s + (0.499 − 0.866i)10-s − 3.77·11-s + (2.38 − 4.13i)13-s + (0.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s + (2.88 − 3.26i)19-s − 0.999·20-s + (1.88 + 3.26i)22-s + (−1.88 + 3.26i)23-s + (−0.499 + 0.866i)25-s − 4.77·26-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.223 + 0.387i)5-s − 0.377·7-s + 0.353·8-s + (0.158 − 0.273i)10-s − 1.13·11-s + (0.661 − 1.14i)13-s + (0.133 + 0.231i)14-s + (−0.125 − 0.216i)16-s + (0.662 − 0.749i)19-s − 0.223·20-s + (0.402 + 0.696i)22-s + (−0.393 + 0.681i)23-s + (−0.0999 + 0.173i)25-s − 0.935·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.805 + 0.592i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.805 + 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1710\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 19\)
Sign: $-0.805 + 0.592i$
Analytic conductor: \(13.6544\)
Root analytic conductor: \(3.69518\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1710} (1261, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1710,\ (\ :1/2),\ -0.805 + 0.592i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7245862046\)
\(L(\frac12)\) \(\approx\) \(0.7245862046\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (-2.88 + 3.26i)T \)
good7 \( 1 + T + 7T^{2} \)
11 \( 1 + 3.77T + 11T^{2} \)
13 \( 1 + (-2.38 + 4.13i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (1.88 - 3.26i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3 - 5.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 2.77T + 31T^{2} \)
37 \( 1 + T + 37T^{2} \)
41 \( 1 + (1.88 + 3.26i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.38 + 7.59i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.88 + 8.46i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.77 + 11.7i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.38 - 2.40i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.38 + 9.32i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.77 + 6.53i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-1.61 - 2.79i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.15 + 8.93i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 6T + 83T^{2} \)
89 \( 1 + (-8.65 + 14.9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5 - 8.66i)T + (-48.5 + 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.142483151203526066914350009104, −8.257593858630796212910817857739, −7.60128769549599167833997974064, −6.73631177240396747573442834584, −5.62607961917952943067613172851, −4.99698836654084967422738563999, −3.48326513611320341990286341542, −3.04867153721466222521826600456, −1.85848547816464928126853671456, −0.31645877489766286185960925369, 1.34322153316599753844086464463, 2.62373501898683998913480696732, 3.96936924774373777192772075324, 4.83527607065806562545660338066, 5.81380327594089397764713232262, 6.35554573889653594451746804921, 7.35026341655802312313486916267, 8.119743453999634349310518524785, 8.732335153358343323592820588462, 9.675558728701782503922152348068

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