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Properties

Label 2-342-171.4-c1-0-17
Degree $2$
Conductor $342$
Sign $-0.897 + 0.440i$
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 + (−0.908 − 1.47i)3-s + (−0.939 + 0.342i)4-s + (−0.355 − 2.01i)5-s + (1.29 − 1.15i)6-s − 1.40·7-s + (−0.5 − 0.866i)8-s + (−1.34 + 2.67i)9-s + (1.92 − 0.700i)10-s + (−2.09 + 3.62i)11-s + (1.35 + 1.07i)12-s + (−1.63 − 1.37i)13-s + (−0.244 − 1.38i)14-s + (−2.65 + 2.35i)15-s + (0.766 − 0.642i)16-s + (−6.80 − 2.47i)17-s + ⋯
L(s)  = 1  + (0.122 + 0.696i)2-s + (−0.524 − 0.851i)3-s + (−0.469 + 0.171i)4-s + (−0.159 − 0.901i)5-s + (0.528 − 0.469i)6-s − 0.531·7-s + (−0.176 − 0.306i)8-s + (−0.449 + 0.893i)9-s + (0.608 − 0.221i)10-s + (−0.630 + 1.09i)11-s + (0.392 + 0.310i)12-s + (−0.453 − 0.380i)13-s + (−0.0652 − 0.370i)14-s + (−0.684 + 0.608i)15-s + (0.191 − 0.160i)16-s + (−1.64 − 0.600i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.897 + 0.440i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.897 + 0.440i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(342\)    =    \(2 \cdot 3^{2} \cdot 19\)
Sign: $-0.897 + 0.440i$
Analytic conductor: \(2.73088\)
Root analytic conductor: \(1.65253\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{342} (175, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 342,\ (\ :1/2),\ -0.897 + 0.440i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0549375 - 0.236905i\)
\(L(\frac12)\) \(\approx\) \(0.0549375 - 0.236905i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.908 + 1.47i)T \)
19 \( 1 + (4.20 - 1.13i)T \)
good5 \( 1 + (0.355 + 2.01i)T + (-4.69 + 1.71i)T^{2} \)
7 \( 1 + 1.40T + 7T^{2} \)
11 \( 1 + (2.09 - 3.62i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.63 + 1.37i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (6.80 + 2.47i)T + (13.0 + 10.9i)T^{2} \)
23 \( 1 + (-3.81 + 1.38i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (7.00 + 5.88i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (-1.84 - 3.19i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 8.68T + 37T^{2} \)
41 \( 1 + (10.1 + 3.68i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (-5.99 - 2.18i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (6.92 + 5.81i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 + (-4.42 - 3.70i)T + (9.20 + 52.1i)T^{2} \)
59 \( 1 + (-8.27 + 6.93i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (-0.196 + 1.11i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (0.666 - 3.78i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (3.85 - 3.23i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (1.09 + 6.22i)T + (-68.5 + 24.9i)T^{2} \)
79 \( 1 + (-1.01 + 0.851i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 - 2.51T + 83T^{2} \)
89 \( 1 + (0.623 - 3.53i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (1.01 + 5.73i)T + (-91.1 + 33.1i)T^{2} \)
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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

−11.33353681103864612208105027629, −10.13154039753915895824257172428, −9.019016152933050699313787443625, −8.120887707678918851310092315897, −7.16729806151445308965290568926, −6.42846073799115323032556514721, −5.18068308353481533950111397747, −4.49339599361620973950492357941, −2.32491810242123678028094470309, −0.16202948442674514828013222592, 2.66857639608043270331043776917, 3.65158984237014558927291388766, 4.76415974965751970895621440606, 6.02089558048033838363823193026, 6.86990919176402158102780770620, 8.546587395524577300220507208028, 9.366376122577546955221254917881, 10.38984655942403773881560427637, 11.10864071146190512141377411848, 11.35336520100689876737996632325

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