↑

Properties

Label 2-182-13.3-c1-0-5
Degree $2$
Conductor $182$
Sign $-0.872 + 0.488i$
Analytic cond. $1.45327$
Root an. cond. $1.20551$
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  + (0.5 + 0.866i)2-s + (−1.5 − 2.59i)3-s + (−0.499 + 0.866i)4-s − 3·5-s + (1.5 − 2.59i)6-s + (−0.5 + 0.866i)7-s − 0.999·8-s + (−3 + 5.19i)9-s + (−1.5 − 2.59i)10-s + (−2 − 3.46i)11-s + 3·12-s + (−2.5 − 2.59i)13-s − 0.999·14-s + (4.5 + 7.79i)15-s + (−0.5 − 0.866i)16-s + (−1 + 1.73i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.866 − 1.49i)3-s + (−0.249 + 0.433i)4-s − 1.34·5-s + (0.612 − 1.06i)6-s + (−0.188 + 0.327i)7-s − 0.353·8-s + (−1 + 1.73i)9-s + (−0.474 − 0.821i)10-s + (−0.603 − 1.04i)11-s + 0.866·12-s + (−0.693 − 0.720i)13-s − 0.267·14-s + (1.16 + 2.01i)15-s + (−0.125 − 0.216i)16-s + (−0.242 + 0.420i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 182 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.872 + 0.488i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 182 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.872 + 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(182\)    =    \(2 \cdot 7 \cdot 13\)
Sign: $-0.872 + 0.488i$
Analytic conductor: \(1.45327\)
Root analytic conductor: \(1.20551\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{182} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 182,\ (\ :1/2),\ -0.872 + 0.488i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0806485 - 0.308893i\)
\(L(\frac12)\) \(\approx\) \(0.0806485 - 0.308893i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (2.5 + 2.59i)T \)
good3 \( 1 + (1.5 + 2.59i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + 3T + 5T^{2} \)
11 \( 1 + (2 + 3.46i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (1 - 1.73i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2 + 3.46i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1 - 1.73i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 10T + 31T^{2} \)
37 \( 1 + (2 + 3.46i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (6 + 10.3i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 10T + 47T^{2} \)
53 \( 1 + 4T + 53T^{2} \)
59 \( 1 + (4.5 - 7.79i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.5 - 11.2i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1 + 1.73i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-7.5 + 12.9i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 2T + 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 + 8T + 83T^{2} \)
89 \( 1 + (7 + 12.1i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1 - 1.73i)T + (-48.5 - 84.0i)T^{2} \)
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

−12.21772241991166314578177409725, −11.70553225241361615318189628006, −10.69435733390401297513650203817, −8.548972606173585746104525253198, −7.80949696043338934290441743590, −7.05549786317044839125294110409, −5.98505289199423513830832531688, −4.95248610359664002307541205395, −3.03127502992163929852536246079, −0.27776965902087070606828979881, 3.28988230344666539676917959792, 4.51160191415912213534975852615, 4.83694691362477461943272633733, 6.60558395677184358337269831294, 8.068777632257990038263725549675, 9.732278836772039130562923954162, 10.05214337212330223479033488631, 11.29267564529598022245634967740, 11.75539898967851983640734450003, 12.57948753755315399991653840085

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