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Properties

Label 2-342-19.9-c1-0-1
Degree $2$
Conductor $342$
Sign $0.537 - 0.843i$
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  + (−0.173 − 0.984i)2-s + (−0.939 + 0.342i)4-s + (−2.20 − 0.802i)5-s + (−1.78 + 3.09i)7-s + (0.5 + 0.866i)8-s + (−0.407 + 2.31i)10-s + (1.35 + 2.35i)11-s + (4.14 + 3.47i)13-s + (3.35 + 1.22i)14-s + (0.766 − 0.642i)16-s + (0.673 + 3.82i)17-s + (1.01 − 4.23i)19-s + 2.34·20-s + (2.08 − 1.74i)22-s + (−7.73 + 2.81i)23-s + ⋯
L(s)  = 1  + (−0.122 − 0.696i)2-s + (−0.469 + 0.171i)4-s + (−0.986 − 0.359i)5-s + (−0.675 + 1.16i)7-s + (0.176 + 0.306i)8-s + (−0.128 + 0.731i)10-s + (0.409 + 0.709i)11-s + (1.14 + 0.964i)13-s + (0.897 + 0.326i)14-s + (0.191 − 0.160i)16-s + (0.163 + 0.926i)17-s + (0.231 − 0.972i)19-s + 0.524·20-s + (0.443 − 0.372i)22-s + (−1.61 + 0.587i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.537 - 0.843i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.537 - 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(342\)    =    \(2 \cdot 3^{2} \cdot 19\)
Sign: $0.537 - 0.843i$
Analytic conductor: \(2.73088\)
Root analytic conductor: \(1.65253\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{342} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 342,\ (\ :1/2),\ 0.537 - 0.843i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.630442 + 0.345817i\)
\(L(\frac12)\) \(\approx\) \(0.630442 + 0.345817i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.173 + 0.984i)T \)
3 \( 1 \)
19 \( 1 + (-1.01 + 4.23i)T \)
good5 \( 1 + (2.20 + 0.802i)T + (3.83 + 3.21i)T^{2} \)
7 \( 1 + (1.78 - 3.09i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.35 - 2.35i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-4.14 - 3.47i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (-0.673 - 3.82i)T + (-15.9 + 5.81i)T^{2} \)
23 \( 1 + (7.73 - 2.81i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (0.613 - 3.47i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (3.26 - 5.65i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 0.389T + 37T^{2} \)
41 \( 1 + (-1.48 + 1.24i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (4.71 + 1.71i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (0.518 - 2.94i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (-7.80 + 2.84i)T + (40.6 - 34.0i)T^{2} \)
59 \( 1 + (0.474 + 2.68i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (5.91 - 2.15i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (2.59 - 14.7i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (-8.47 - 3.08i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (-7.88 + 6.61i)T + (12.6 - 71.8i)T^{2} \)
79 \( 1 + (-9.96 + 8.36i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (4.08 - 7.07i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (8.98 + 7.53i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (1.49 + 8.47i)T + (-91.1 + 33.1i)T^{2} \)
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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

−11.87752368150156435537153697013, −10.94927630824832487830665197739, −9.719527628008944969223148433946, −8.921936413646966198519046642294, −8.279681917837922681625139406614, −6.91072176238179274080979733642, −5.73920643708574912411784209532, −4.30930873915252657092767137935, −3.46397862724651728155624065568, −1.83548845539946204741832477608, 0.53969608439179075720649393659, 3.50461711698449477912898286814, 4.00867561787498430431935763851, 5.76672815024368492931374863826, 6.60580334052414605165076281544, 7.73967798711097620145459588584, 8.125841802965782560783515855994, 9.536608754321642892825602767393, 10.42121541413079674178638831919, 11.26022905135273221112667752939

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