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Properties

Label 2-342-57.50-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

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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: $\chi_{342} (107, \cdot )$
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 \( 1 + (16.3 - 9.43i)T + (48.5 - 84.0i)T^{2} \)
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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

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

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