↑

Properties

Label 2-342-171.164-c1-0-10
Degree $2$
Conductor $342$
Sign $0.790 - 0.612i$
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  + 2-s + 1.73i·3-s + 4-s + (1.5 + 0.866i)5-s + 1.73i·6-s + (2.5 − 4.33i)7-s + 8-s − 2.99·9-s + (1.5 + 0.866i)10-s + (1.5 + 0.866i)11-s + 1.73i·12-s + (2.5 − 4.33i)14-s + (−1.49 + 2.59i)15-s + 16-s + (−4.5 + 2.59i)17-s − 2.99·18-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.999i·3-s + 0.5·4-s + (0.670 + 0.387i)5-s + 0.707i·6-s + (0.944 − 1.63i)7-s + 0.353·8-s − 0.999·9-s + (0.474 + 0.273i)10-s + (0.452 + 0.261i)11-s + 0.499i·12-s + (0.668 − 1.15i)14-s + (−0.387 + 0.670i)15-s + 0.250·16-s + (−1.09 + 0.630i)17-s − 0.707·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.790 - 0.612i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.790 - 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(342\)    =    \(2 \cdot 3^{2} \cdot 19\)
Sign: $0.790 - 0.612i$
Analytic conductor: \(2.73088\)
Root analytic conductor: \(1.65253\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{342} (335, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 342,\ (\ :1/2),\ 0.790 - 0.612i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.14125 + 0.733209i\)
\(L(\frac12)\) \(\approx\) \(2.14125 + 0.733209i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 - 1.73iT \)
19 \( 1 + (4 - 1.73i)T \)
good5 \( 1 + (-1.5 - 0.866i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (-2.5 + 4.33i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.5 - 0.866i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + (4.5 - 2.59i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 - 3.46iT - 23T^{2} \)
29 \( 1 + (1.5 + 2.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (7.5 - 4.33i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + (-4.5 + 7.79i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
47 \( 1 + (-4.5 + 2.59i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.5 - 2.59i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.5 - 2.59i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.5 + 6.06i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + 3.46iT - 67T^{2} \)
71 \( 1 + (-7.5 - 12.9i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (5.5 + 9.52i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 - 10.3iT - 79T^{2} \)
83 \( 1 + (4.5 + 2.59i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (7.5 - 12.9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 6.92iT - 97T^{2} \)
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

−11.22899330086694904151886502534, −10.78738958477716485982067829108, −10.14353477856286243217911575881, −8.964142032392409777549529652875, −7.70811063350802033608640318794, −6.64515299429398199210759436481, −5.55585872312141876337184873541, −4.30866884168892508591816402226, −3.87180823064490578489902619684, −2.01673887893303704352032318001, 1.83015243557388331147581833272, 2.59706972886838760091335730702, 4.65704699844454830516419436111, 5.68000958827942055263278352782, 6.26168109372129724993529472621, 7.52335777512032103704048685984, 8.747007412055489591390362117234, 9.130404913232230321908438769614, 11.13042247309127978389712776015, 11.48207895479821921997361959057

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