↑

Properties

Label 2-342-171.110-c1-0-17
Degree $2$
Conductor $342$
Sign $-0.671 + 0.741i$
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 + (−0.0593 − 1.73i)3-s + (0.173 − 0.984i)4-s + (0.784 − 2.15i)5-s + (1.15 + 1.28i)6-s + (−1.44 − 2.50i)7-s + (0.500 + 0.866i)8-s + (−2.99 + 0.205i)9-s + (0.784 + 2.15i)10-s + 1.23i·11-s + (−1.71 − 0.242i)12-s + (−0.345 − 0.950i)13-s + (2.71 + 0.989i)14-s + (−3.77 − 1.22i)15-s + (−0.939 − 0.342i)16-s + (−1.81 + 4.99i)17-s + ⋯
L(s)  = 1  + (−0.541 + 0.454i)2-s + (−0.0342 − 0.999i)3-s + (0.0868 − 0.492i)4-s + (0.350 − 0.963i)5-s + (0.472 + 0.525i)6-s + (−0.546 − 0.946i)7-s + (0.176 + 0.306i)8-s + (−0.997 + 0.0684i)9-s + (0.248 + 0.681i)10-s + 0.373i·11-s + (−0.495 − 0.0699i)12-s + (−0.0959 − 0.263i)13-s + (0.726 + 0.264i)14-s + (−0.975 − 0.317i)15-s + (−0.234 − 0.0855i)16-s + (−0.441 + 1.21i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.671 + 0.741i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.671 + 0.741i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(342\)    =    \(2 \cdot 3^{2} \cdot 19\)
Sign: $-0.671 + 0.741i$
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: \((2,\ 342,\ (\ :1/2),\ -0.671 + 0.741i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.308754 - 0.696082i\)
\(L(\frac12)\) \(\approx\) \(0.308754 - 0.696082i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.0593 + 1.73i)T \)
19 \( 1 + (4.11 + 1.43i)T \)
good5 \( 1 + (-0.784 + 2.15i)T + (-3.83 - 3.21i)T^{2} \)
7 \( 1 + (1.44 + 2.50i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 - 1.23iT - 11T^{2} \)
13 \( 1 + (0.345 + 0.950i)T + (-9.95 + 8.35i)T^{2} \)
17 \( 1 + (1.81 - 4.99i)T + (-13.0 - 10.9i)T^{2} \)
23 \( 1 + (-2.46 - 0.435i)T + (21.6 + 7.86i)T^{2} \)
29 \( 1 + (-0.832 + 4.72i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + 10.0iT - 31T^{2} \)
37 \( 1 + 2.65iT - 37T^{2} \)
41 \( 1 + (2.59 - 2.17i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (1.64 + 9.35i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (-9.52 - 1.67i)T + (44.1 + 16.0i)T^{2} \)
53 \( 1 + (-4.95 - 4.15i)T + (9.20 + 52.1i)T^{2} \)
59 \( 1 + (2.27 + 12.9i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-5.05 + 1.84i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (4.46 - 5.32i)T + (-11.6 - 65.9i)T^{2} \)
71 \( 1 + (6.14 - 5.15i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (-1.33 - 7.59i)T + (-68.5 + 24.9i)T^{2} \)
79 \( 1 + (-2.44 + 6.72i)T + (-60.5 - 50.7i)T^{2} \)
83 \( 1 + (-0.460 + 0.265i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (0.359 - 2.04i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (-7.87 - 9.38i)T + (-16.8 + 95.5i)T^{2} \)
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

−11.07289023537331687272455327171, −10.16190424897967828010464549994, −9.096655159044934936207755124058, −8.320450577100686276154822338729, −7.37718926629389466036266417015, −6.50668509507394820548197278365, −5.59894973876207053349260717293, −4.17456513050023874163418201374, −2.08587180280859276809839678629, −0.61750272162108336048231045973, 2.56575319361584467655067292597, 3.25696265917078809823472540556, 4.82461670719095631151598034503, 6.09692669029242052214234502605, 7.01831711099370829158979596441, 8.696195535696026036382985472162, 9.080906734034080089009655502149, 10.17215895125267485011610288248, 10.70032599148520002846924840104, 11.59377334644220985349182615444

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