↑

Properties

Label 2-342-171.110-c1-0-12
Degree $2$
Conductor $342$
Sign $-0.256 + 0.966i$
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  + (0.766 − 0.642i)2-s + (−1.73 + 0.0369i)3-s + (0.173 − 0.984i)4-s + (0.303 − 0.833i)5-s + (−1.30 + 1.14i)6-s + (0.619 + 1.07i)7-s + (−0.500 − 0.866i)8-s + (2.99 − 0.127i)9-s + (−0.303 − 0.833i)10-s − 4.47i·11-s + (−0.264 + 1.71i)12-s + (−1.65 − 4.55i)13-s + (1.16 + 0.423i)14-s + (−0.494 + 1.45i)15-s + (−0.939 − 0.342i)16-s + (1.16 − 3.20i)17-s + ⋯
L(s)  = 1  + (0.541 − 0.454i)2-s + (−0.999 + 0.0213i)3-s + (0.0868 − 0.492i)4-s + (0.135 − 0.372i)5-s + (−0.531 + 0.465i)6-s + (0.234 + 0.405i)7-s + (−0.176 − 0.306i)8-s + (0.999 − 0.0426i)9-s + (−0.0959 − 0.263i)10-s − 1.34i·11-s + (−0.0763 + 0.494i)12-s + (−0.460 − 1.26i)13-s + (0.311 + 0.113i)14-s + (−0.127 + 0.375i)15-s + (−0.234 − 0.0855i)16-s + (0.283 − 0.778i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.256 + 0.966i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.256 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(342\)    =    \(2 \cdot 3^{2} \cdot 19\)
Sign: $-0.256 + 0.966i$
Analytic conductor: \(2.73088\)
Root analytic conductor: \(1.65253\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{342} (281, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 342,\ (\ :1/2),\ -0.256 + 0.966i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.754801 - 0.980982i\)
\(L(\frac12)\) \(\approx\) \(0.754801 - 0.980982i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.766 + 0.642i)T \)
3 \( 1 + (1.73 - 0.0369i)T \)
19 \( 1 + (4.29 - 0.727i)T \)
good5 \( 1 + (-0.303 + 0.833i)T + (-3.83 - 3.21i)T^{2} \)
7 \( 1 + (-0.619 - 1.07i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + 4.47iT - 11T^{2} \)
13 \( 1 + (1.65 + 4.55i)T + (-9.95 + 8.35i)T^{2} \)
17 \( 1 + (-1.16 + 3.20i)T + (-13.0 - 10.9i)T^{2} \)
23 \( 1 + (-0.542 - 0.0956i)T + (21.6 + 7.86i)T^{2} \)
29 \( 1 + (0.188 - 1.06i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + 3.17iT - 31T^{2} \)
37 \( 1 + 0.210iT - 37T^{2} \)
41 \( 1 + (-6.75 + 5.66i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (-1.98 - 11.2i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (-9.59 - 1.69i)T + (44.1 + 16.0i)T^{2} \)
53 \( 1 + (-4.72 - 3.96i)T + (9.20 + 52.1i)T^{2} \)
59 \( 1 + (0.254 + 1.44i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (4.88 - 1.77i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (8.44 - 10.0i)T + (-11.6 - 65.9i)T^{2} \)
71 \( 1 + (-0.567 + 0.475i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (-1.49 - 8.50i)T + (-68.5 + 24.9i)T^{2} \)
79 \( 1 + (2.86 - 7.87i)T + (-60.5 - 50.7i)T^{2} \)
83 \( 1 + (6.65 - 3.83i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (1.80 - 10.2i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (2.59 + 3.08i)T + (-16.8 + 95.5i)T^{2} \)
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

−11.19878912334902660042744649005, −10.72962562209210723829356463321, −9.656551872846059368282042889932, −8.537620850895248338135743699194, −7.26203941877969855141630765960, −5.82475756446349036126011999743, −5.50254889221114026379031632346, −4.30511021752683136821735643577, −2.80297237002817225626371355788, −0.857773105106758020141451249002, 2.03315031494268954719247389185, 4.17328642751774428705810032296, 4.72601897927607586942071909861, 6.06872701436111530699679096271, 6.88694604268141656713635459538, 7.50954172904744899841684372494, 9.042569750811799326002497770188, 10.24582385307207467269606532981, 10.86198425963228915488867442481, 12.07892818246654552460303014222

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