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Properties

Label 2-342-19.4-c1-0-8
Degree $2$
Conductor $342$
Sign $0.998 + 0.0463i$
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  + (0.939 + 0.342i)2-s + (0.766 + 0.642i)4-s + (2.97 − 2.49i)5-s + (−0.613 + 1.06i)7-s + (0.500 + 0.866i)8-s + (3.64 − 1.32i)10-s + (−1.06 − 1.83i)11-s + (0.0851 − 0.482i)13-s + (−0.939 + 0.788i)14-s + (0.173 + 0.984i)16-s + (−5.19 − 1.89i)17-s + (2.77 + 3.35i)19-s + 3.87·20-s + (−0.368 − 2.08i)22-s + (6.85 + 5.74i)23-s + ⋯
L(s)  = 1  + (0.664 + 0.241i)2-s + (0.383 + 0.321i)4-s + (1.32 − 1.11i)5-s + (−0.231 + 0.401i)7-s + (0.176 + 0.306i)8-s + (1.15 − 0.419i)10-s + (−0.319 − 0.553i)11-s + (0.0236 − 0.133i)13-s + (−0.251 + 0.210i)14-s + (0.0434 + 0.246i)16-s + (−1.26 − 0.458i)17-s + (0.637 + 0.770i)19-s + 0.867·20-s + (−0.0785 − 0.445i)22-s + (1.42 + 1.19i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0463i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0463i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(342\)    =    \(2 \cdot 3^{2} \cdot 19\)
Sign: $0.998 + 0.0463i$
Analytic conductor: \(2.73088\)
Root analytic conductor: \(1.65253\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{342} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 342,\ (\ :1/2),\ 0.998 + 0.0463i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.22262 - 0.0515016i\)
\(L(\frac12)\) \(\approx\) \(2.22262 - 0.0515016i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.939 - 0.342i)T \)
3 \( 1 \)
19 \( 1 + (-2.77 - 3.35i)T \)
good5 \( 1 + (-2.97 + 2.49i)T + (0.868 - 4.92i)T^{2} \)
7 \( 1 + (0.613 - 1.06i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.06 + 1.83i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.0851 + 0.482i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (5.19 + 1.89i)T + (13.0 + 10.9i)T^{2} \)
23 \( 1 + (-6.85 - 5.74i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (7.96 - 2.89i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (1.20 - 2.08i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 1.69T + 37T^{2} \)
41 \( 1 + (0.277 + 1.57i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (5.08 - 4.26i)T + (7.46 - 42.3i)T^{2} \)
47 \( 1 + (-2.03 + 0.742i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 + (6.80 + 5.71i)T + (9.20 + 52.1i)T^{2} \)
59 \( 1 + (10.7 + 3.90i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-0.0320 - 0.0269i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (4.20 - 1.53i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (2.02 - 1.69i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (2.66 + 15.1i)T + (-68.5 + 24.9i)T^{2} \)
79 \( 1 + (0.809 + 4.58i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (-6.24 + 10.8i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (1.46 - 8.32i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (-13.5 - 4.91i)T + (74.3 + 62.3i)T^{2} \)
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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

−11.68698257424471831029835900242, −10.68306305123415864002897344009, −9.335777619829122968051693041712, −9.007696220259026302288710940841, −7.65871106601629432921209182381, −6.33866744826431810762125067640, −5.49750475854216018732727996825, −4.88074047955053119635819148619, −3.19046423039792206033741577423, −1.71625126830683767274903728242, 2.05415999510673551794126906891, 3.00046479096867885549956718289, 4.49743158354311285248120205491, 5.69364992419396184046176598801, 6.66766004396014390914010500966, 7.23182913258857538994337594409, 9.065109807362815408806319160945, 9.896798879975463926860934019182, 10.76488452361835997241303519833, 11.24502189711672369455982111496

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