↑

Properties

Label 2-342-171.59-c1-0-18
Degree $2$
Conductor $342$
Sign $-0.0269 + 0.999i$
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 + (0.809 − 1.53i)3-s + (0.766 − 0.642i)4-s + (−1.35 − 0.239i)5-s + (0.237 − 1.71i)6-s + (−0.687 + 1.19i)7-s + (0.500 − 0.866i)8-s + (−1.68 − 2.47i)9-s + (−1.35 + 0.239i)10-s − 6.17i·11-s + (−0.363 − 1.69i)12-s + (3.78 − 0.667i)13-s + (−0.238 + 1.35i)14-s + (−1.46 + 1.88i)15-s + (0.173 − 0.984i)16-s + (4.09 + 0.722i)17-s + ⋯
L(s)  = 1  + (0.664 − 0.241i)2-s + (0.467 − 0.883i)3-s + (0.383 − 0.321i)4-s + (−0.606 − 0.106i)5-s + (0.0969 − 0.700i)6-s + (−0.259 + 0.450i)7-s + (0.176 − 0.306i)8-s + (−0.562 − 0.826i)9-s + (−0.429 + 0.0756i)10-s − 1.86i·11-s + (−0.104 − 0.488i)12-s + (1.04 − 0.185i)13-s + (−0.0638 + 0.361i)14-s + (−0.378 + 0.486i)15-s + (0.0434 − 0.246i)16-s + (0.993 + 0.175i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0269 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0269 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(342\)    =    \(2 \cdot 3^{2} \cdot 19\)
Sign: $-0.0269 + 0.999i$
Analytic conductor: \(2.73088\)
Root analytic conductor: \(1.65253\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{342} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 342,\ (\ :1/2),\ -0.0269 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.38474 - 1.42265i\)
\(L(\frac12)\) \(\approx\) \(1.38474 - 1.42265i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.809 + 1.53i)T \)
19 \( 1 + (1.80 - 3.96i)T \)
good5 \( 1 + (1.35 + 0.239i)T + (4.69 + 1.71i)T^{2} \)
7 \( 1 + (0.687 - 1.19i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + 6.17iT - 11T^{2} \)
13 \( 1 + (-3.78 + 0.667i)T + (12.2 - 4.44i)T^{2} \)
17 \( 1 + (-4.09 - 0.722i)T + (15.9 + 5.81i)T^{2} \)
23 \( 1 + (-3.37 - 4.02i)T + (-3.99 + 22.6i)T^{2} \)
29 \( 1 + (4.02 - 3.38i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 - 8.32iT - 31T^{2} \)
37 \( 1 - 0.963iT - 37T^{2} \)
41 \( 1 + (-8.84 + 3.21i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (-5.12 - 4.30i)T + (7.46 + 42.3i)T^{2} \)
47 \( 1 + (-2.86 - 3.41i)T + (-8.16 + 46.2i)T^{2} \)
53 \( 1 + (-0.893 - 0.325i)T + (40.6 + 34.0i)T^{2} \)
59 \( 1 + (7.63 + 6.40i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (1.74 + 9.88i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (-1.22 + 3.37i)T + (-51.3 - 43.0i)T^{2} \)
71 \( 1 + (-2.80 + 1.02i)T + (54.3 - 45.6i)T^{2} \)
73 \( 1 + (-5.16 - 4.33i)T + (12.6 + 71.8i)T^{2} \)
79 \( 1 + (13.0 + 2.30i)T + (74.2 + 27.0i)T^{2} \)
83 \( 1 + (14.8 + 8.56i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-3.67 + 3.08i)T + (15.4 - 87.6i)T^{2} \)
97 \( 1 + (-4.23 - 11.6i)T + (-74.3 + 62.3i)T^{2} \)
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

−11.43022420490403844599594418137, −10.80065389951897569811283954441, −9.189587008612571394367522194692, −8.362482062890975763307074342953, −7.57157898842259285685987659163, −6.14095211333476292680774101697, −5.69667240690553323626185262710, −3.65752009895425755040847968756, −3.13088898720558180615887660406, −1.23889589884306320198436654168, 2.51021513281652449010688528687, 3.96615591396049357591316098760, 4.36617739750606607523787508560, 5.71069605072160902420114382009, 7.12807682496310914824394122330, 7.78334066525595434932333979300, 9.044163941242044492958961209837, 9.940459118790522774718559055789, 10.88184248700362566774204508892, 11.71190538787147436168597650934

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