Properties

Label 2-342-19.4-c1-0-0
Degree $2$
Conductor $342$
Sign $0.290 - 0.956i$
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  + (−0.939 − 0.342i)2-s + (0.766 + 0.642i)4-s + (−0.907 + 0.761i)5-s + (−0.266 + 0.460i)7-s + (−0.500 − 0.866i)8-s + (1.11 − 0.405i)10-s + (0.939 + 1.62i)11-s + (−0.673 + 3.82i)13-s + (0.407 − 0.342i)14-s + (0.173 + 0.984i)16-s + (1.09 + 0.397i)17-s + (−3.93 + 1.86i)19-s − 1.18·20-s + (−0.326 − 1.85i)22-s + (5.13 + 4.30i)23-s + ⋯
L(s)  = 1  + (−0.664 − 0.241i)2-s + (0.383 + 0.321i)4-s + (−0.405 + 0.340i)5-s + (−0.100 + 0.174i)7-s + (−0.176 − 0.306i)8-s + (0.352 − 0.128i)10-s + (0.283 + 0.490i)11-s + (−0.186 + 1.05i)13-s + (0.108 − 0.0914i)14-s + (0.0434 + 0.246i)16-s + (0.264 + 0.0964i)17-s + (−0.903 + 0.427i)19-s − 0.264·20-s + (−0.0695 − 0.394i)22-s + (1.07 + 0.898i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.290 - 0.956i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.290 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(342\)    =    \(2 \cdot 3^{2} \cdot 19\)
Sign: $0.290 - 0.956i$
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: \((2,\ 342,\ (\ :1/2),\ 0.290 - 0.956i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.617168 + 0.457524i\)
\(L(\frac12)\) \(\approx\) \(0.617168 + 0.457524i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.93 - 1.86i)T \)
good5 \( 1 + (0.907 - 0.761i)T + (0.868 - 4.92i)T^{2} \)
7 \( 1 + (0.266 - 0.460i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.939 - 1.62i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.673 - 3.82i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (-1.09 - 0.397i)T + (13.0 + 10.9i)T^{2} \)
23 \( 1 + (-5.13 - 4.30i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (3.77 - 1.37i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (-0.979 + 1.69i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 6.88T + 37T^{2} \)
41 \( 1 + (-1.56 - 8.84i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (-1.85 + 1.55i)T + (7.46 - 42.3i)T^{2} \)
47 \( 1 + (-1.91 + 0.698i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 + (9.93 + 8.33i)T + (9.20 + 52.1i)T^{2} \)
59 \( 1 + (2.51 + 0.916i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (8.69 + 7.29i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (-10.4 + 3.82i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (4.65 - 3.90i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (0.0569 + 0.322i)T + (-68.5 + 24.9i)T^{2} \)
79 \( 1 + (2.80 + 15.8i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (-5.78 + 10.0i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-0.618 + 3.50i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (5.52 + 2.01i)T + (74.3 + 62.3i)T^{2} \)
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

−11.46622791294043423028655597667, −10.94083224341568034219716458687, −9.662779150571269271153206565212, −9.161824514103627883095826014089, −7.920290450046927756293201119844, −7.13229152019743198425231224686, −6.13598886406569880410886546204, −4.53084465031989418325418703523, −3.30151892042349132807794789110, −1.78281586303432663074425141247, 0.68049974597636029503536327717, 2.73820375678545408775119331576, 4.27120015667433757236664875066, 5.57404651198617398590633958541, 6.65109169167062413111174491058, 7.71755720401888061470832883572, 8.501937607421933206818886994404, 9.330172309758018413397604268116, 10.48422178237886763482541179304, 11.06336955884862509222158481491

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