↑

Properties

Label 2-342-57.8-c1-0-2
Degree $2$
Conductor $342$
Sign $0.600 + 0.799i$
Analytic cond. $2.73088$
Root an. cond. $1.65253$
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 + (1.22 + 0.707i)5-s + 1.44·7-s − 0.999·8-s + (1.22 − 0.707i)10-s − 0.635i·11-s + (5.17 − 2.98i)13-s + (0.724 − 1.25i)14-s + (−0.5 + 0.866i)16-s + (−1.77 − 1.02i)17-s + (4 + 1.73i)19-s − 1.41i·20-s + (−0.550 − 0.317i)22-s + (−4.89 + 2.82i)23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.547 + 0.316i)5-s + 0.547·7-s − 0.353·8-s + (0.387 − 0.223i)10-s − 0.191i·11-s + (1.43 − 0.828i)13-s + (0.193 − 0.335i)14-s + (−0.125 + 0.216i)16-s + (−0.430 − 0.248i)17-s + (0.917 + 0.397i)19-s − 0.316i·20-s + (−0.117 − 0.0677i)22-s + (−1.02 + 0.589i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.600 + 0.799i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.600 + 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(342\)    =    \(2 \cdot 3^{2} \cdot 19\)
Sign: $0.600 + 0.799i$
Analytic conductor: \(2.73088\)
Root analytic conductor: \(1.65253\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{342} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 342,\ (\ :1/2),\ 0.600 + 0.799i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.59995 - 0.798910i\)
\(L(\frac12)\) \(\approx\) \(1.59995 - 0.798910i\)
\(L(\frac{3}{2})\) 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 \( 1 \)
19 \( 1 + (-4 - 1.73i)T \)
good5 \( 1 + (-1.22 - 0.707i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 - 1.44T + 7T^{2} \)
11 \( 1 + 0.635iT - 11T^{2} \)
13 \( 1 + (-5.17 + 2.98i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.77 + 1.02i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (4.89 - 2.82i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.22 + 2.12i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 0.953iT - 31T^{2} \)
37 \( 1 - 2.51iT - 37T^{2} \)
41 \( 1 + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.94 - 3.37i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.77 + 1.02i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.44 - 4.24i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (7.22 - 12.5i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.72 - 2.98i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (8.84 - 5.10i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (3 - 5.19i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-1.05 + 1.81i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.825 - 0.476i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 11.8iT - 83T^{2} \)
89 \( 1 + (-6.12 - 10.6i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (16.3 + 9.43i)T + (48.5 + 84.0i)T^{2} \)
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

−11.38669572473492614031300756532, −10.55871279377759065033888524295, −9.806977431495908232875507873763, −8.670917967417963822949830942190, −7.71289871846927027415641561069, −6.19128663714793441561582383377, −5.53240496602713295945333750228, −4.13722927605505089241139764573, −2.95640398964277002254706102000, −1.47386393601436772054816722080, 1.78508797749065650417287367966, 3.68481166797119784202498181439, 4.79520365470205555239280520475, 5.83233843612459099147996027573, 6.70121054443875997899314226758, 7.88515924101842551361510319302, 8.789892934839718192460779024492, 9.565514757801509182999223335513, 10.88699967151691558751103988852, 11.68212474830297006962558459892

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