Properties

Label 2-153-153.106-c1-0-0
Degree $2$
Conductor $153$
Sign $-0.0732 - 0.997i$
Analytic cond. $1.22171$
Root an. cond. $1.10531$
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  + (−1.19 − 0.688i)2-s + (−1.71 − 0.266i)3-s + (−0.0512 − 0.0887i)4-s + (−0.723 − 0.193i)5-s + (1.85 + 1.49i)6-s + (−0.511 + 0.137i)7-s + 2.89i·8-s + (2.85 + 0.912i)9-s + (0.729 + 0.729i)10-s + (−4.56 + 1.22i)11-s + (0.0640 + 0.165i)12-s + (3.08 + 5.33i)13-s + (0.705 + 0.188i)14-s + (1.18 + 0.524i)15-s + (1.89 − 3.27i)16-s + (−3.76 + 1.67i)17-s + ⋯
L(s)  = 1  + (−0.843 − 0.487i)2-s + (−0.988 − 0.153i)3-s + (−0.0256 − 0.0443i)4-s + (−0.323 − 0.0866i)5-s + (0.758 + 0.611i)6-s + (−0.193 + 0.0518i)7-s + 1.02i·8-s + (0.952 + 0.304i)9-s + (0.230 + 0.230i)10-s + (−1.37 + 0.368i)11-s + (0.0184 + 0.0478i)12-s + (0.854 + 1.48i)13-s + (0.188 + 0.0505i)14-s + (0.306 + 0.135i)15-s + (0.473 − 0.819i)16-s + (−0.913 + 0.405i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0732 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0732 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(153\)    =    \(3^{2} \cdot 17\)
Sign: $-0.0732 - 0.997i$
Analytic conductor: \(1.22171\)
Root analytic conductor: \(1.10531\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{153} (106, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 153,\ (\ :1/2),\ -0.0732 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.115676 + 0.124489i\)
\(L(\frac12)\) \(\approx\) \(0.115676 + 0.124489i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.71 + 0.266i)T \)
17 \( 1 + (3.76 - 1.67i)T \)
good2 \( 1 + (1.19 + 0.688i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (0.723 + 0.193i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (0.511 - 0.137i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (4.56 - 1.22i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (-3.08 - 5.33i)T + (-6.5 + 11.2i)T^{2} \)
19 \( 1 + 0.0849iT - 19T^{2} \)
23 \( 1 + (0.0237 - 0.0885i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (-0.846 - 3.15i)T + (-25.1 + 14.5i)T^{2} \)
31 \( 1 + (3.83 + 1.02i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (4.82 - 4.82i)T - 37iT^{2} \)
41 \( 1 + (-2.97 + 11.1i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (3.46 + 1.99i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.60 - 4.51i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 8.93iT - 53T^{2} \)
59 \( 1 + (0.484 - 0.279i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.35 + 1.43i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (-5.49 - 9.51i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3.10 + 3.10i)T - 71iT^{2} \)
73 \( 1 + (0.458 - 0.458i)T - 73iT^{2} \)
79 \( 1 + (8.22 - 2.20i)T + (68.4 - 39.5i)T^{2} \)
83 \( 1 + (0.713 + 0.411i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 5.20T + 89T^{2} \)
97 \( 1 + (-2.21 - 8.27i)T + (-84.0 + 48.5i)T^{2} \)
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

−13.00748207351889498542885984823, −11.91201864750661501851128649191, −11.04467903948567109898234308715, −10.42677644534302290815641707655, −9.314216693904923239617767050343, −8.221404694530139444419819268866, −6.91646788488747831809254843936, −5.62999235944377612137000020036, −4.40211713519639378397449973491, −1.89573872922000240423523470220, 0.23970958096882867020029739507, 3.54563351209259667603218833513, 5.19380054436503772752351196192, 6.37573271012995959821129529080, 7.57707313876873536582867483854, 8.359623340661973734061278801695, 9.748390310641259521982323350405, 10.58922159913046428083977020872, 11.44874075336685098400057886811, 12.97678300596701731931926014901

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