↑

Properties

Label 2-342-19.16-c1-0-2
Degree $2$
Conductor $342$
Sign $-0.174 - 0.984i$
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 + (0.173 + 0.984i)4-s + (−0.386 + 2.19i)5-s + (0.326 + 0.565i)7-s + (−0.500 + 0.866i)8-s + (−1.70 + 1.43i)10-s + (−0.766 + 1.32i)11-s + (0.439 + 0.160i)13-s + (−0.113 + 0.642i)14-s + (−0.939 + 0.342i)16-s + (1.61 + 1.35i)17-s + (−2.23 + 3.74i)19-s − 2.22·20-s + (−1.43 + 0.524i)22-s + (−1.02 − 5.81i)23-s + ⋯
L(s)  = 1  + (0.541 + 0.454i)2-s + (0.0868 + 0.492i)4-s + (−0.172 + 0.980i)5-s + (0.123 + 0.213i)7-s + (−0.176 + 0.306i)8-s + (−0.539 + 0.452i)10-s + (−0.230 + 0.400i)11-s + (0.121 + 0.0443i)13-s + (−0.0302 + 0.171i)14-s + (−0.234 + 0.0855i)16-s + (0.391 + 0.328i)17-s + (−0.512 + 0.858i)19-s − 0.497·20-s + (−0.306 + 0.111i)22-s + (−0.213 − 1.21i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.174 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.174 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(342\)    =    \(2 \cdot 3^{2} \cdot 19\)
Sign: $-0.174 - 0.984i$
Analytic conductor: \(2.73088\)
Root analytic conductor: \(1.65253\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{342} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 342,\ (\ :1/2),\ -0.174 - 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.06451 + 1.26948i\)
\(L(\frac12)\) \(\approx\) \(1.06451 + 1.26948i\)
\(L(\frac{3}{2})\) 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 \( 1 \)
19 \( 1 + (2.23 - 3.74i)T \)
good5 \( 1 + (0.386 - 2.19i)T + (-4.69 - 1.71i)T^{2} \)
7 \( 1 + (-0.326 - 0.565i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.766 - 1.32i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.439 - 0.160i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (-1.61 - 1.35i)T + (2.95 + 16.7i)T^{2} \)
23 \( 1 + (1.02 + 5.81i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (-6.38 + 5.35i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (-4.31 - 7.48i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 4.67T + 37T^{2} \)
41 \( 1 + (-3.26 + 1.18i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (-1.78 + 10.1i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-3.55 + 2.98i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 + (-2.07 - 11.7i)T + (-49.8 + 18.1i)T^{2} \)
59 \( 1 + (10.9 + 9.14i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (1.58 + 8.98i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (-0.190 + 0.160i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (-0.772 + 4.38i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (-8.54 + 3.10i)T + (55.9 - 46.9i)T^{2} \)
79 \( 1 + (11.3 - 4.12i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (-1.85 - 3.21i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-15.7 - 5.73i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (-1.89 - 1.58i)T + (16.8 + 95.5i)T^{2} \)
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

−12.11892065995438767810641001836, −10.72138688023275252785921192048, −10.26212166560161696657674275883, −8.728630843977276509007171762055, −7.87203137188075627340885718789, −6.83075112512911465577377307210, −6.10796166259082698360548086159, −4.82375900805252965668911607092, −3.63799032788958837772659139287, −2.40841523266373241746775448818, 1.07103350520553987609692158879, 2.86763085219928917472373215569, 4.25898558503308983074349452777, 5.07955866646192137026524200056, 6.15748847181317843846407196220, 7.51163122662994352355791309935, 8.574030094301035539725334689077, 9.444521778508844519326334935548, 10.51610357966774128939934320715, 11.42334592867188069474478461943

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