↑

Properties

Label 2-1710-19.11-c1-0-8
Degree $2$
Conductor $1710$
Sign $0.658 - 0.752i$
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 + 3·11-s + (−1 − 1.73i)13-s + (−0.5 + 0.866i)14-s + (−0.5 + 0.866i)16-s + (−3 + 5.19i)17-s + (−3.5 + 2.59i)19-s + 0.999·20-s + (1.5 − 2.59i)22-s + (4.5 + 7.79i)23-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 + 0.904·11-s + (−0.277 − 0.480i)13-s + (−0.133 + 0.231i)14-s + (−0.125 + 0.216i)16-s + (−0.727 + 1.26i)17-s + (−0.802 + 0.596i)19-s + 0.223·20-s + (0.319 − 0.553i)22-s + (0.938 + 1.62i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.658 - 0.752i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.658 - 0.752i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1710\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 19\)
Sign: $0.658 - 0.752i$
Analytic conductor: \(13.6544\)
Root analytic conductor: \(3.69518\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1710} (1531, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1710,\ (\ :1/2),\ 0.658 - 0.752i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.336408750\)
\(L(\frac12)\) \(\approx\) \(1.336408750\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.5 - 2.59i)T \)
good7 \( 1 + T + 7T^{2} \)
11 \( 1 - 3T + 11T^{2} \)
13 \( 1 + (1 + 1.73i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (3 - 5.19i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-4.5 - 7.79i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 - 5T + 37T^{2} \)
41 \( 1 + (1.5 - 2.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5 + 8.66i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.5 - 2.59i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (6 - 10.3i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5 - 8.66i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5 - 8.66i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-2 + 3.46i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (7 - 12.1i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 18T + 83T^{2} \)
89 \( 1 + (-7.5 - 12.9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (4 - 6.92i)T + (-48.5 - 84.0i)T^{2} \)
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

−9.474248030723274036933124023360, −8.862633861183675821626943481547, −7.87681680899088239758507011769, −6.92243457205358513387163826538, −6.16420408652214771700532667428, −5.36783902406825406690810595917, −4.08465295734707262378041037337, −3.66004157959015855203132114914, −2.51501277548880281940353046911, −1.36433030448518563543418492150, 0.46459806587223489994971596987, 2.28111575988886751847433447476, 3.40884539147433637376264635329, 4.58221270477119690529292363459, 4.81806255509316413449736356982, 6.28236251870398717486321754110, 6.67478561385550140916300161683, 7.46931627856557785847437982474, 8.513056891460069452039407186177, 9.133403481605266037003026980983

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