↑

Properties

Label 2-306-153.4-c1-0-10
Degree $2$
Conductor $306$
Sign $0.883 + 0.469i$
Analytic cond. $2.44342$
Root an. cond. $1.56314$
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  + (0.866 − 0.5i)2-s + (1.5 − 0.866i)3-s + (0.499 − 0.866i)4-s + (0.732 + 2.73i)5-s + (0.866 − 1.5i)6-s + (−0.267 + i)7-s − 0.999i·8-s + (1.5 − 2.59i)9-s + (2 + 1.99i)10-s + (0.598 − 2.23i)11-s − 1.73i·12-s + (−1.73 + 3i)13-s + (0.267 + i)14-s + (3.46 + 3.46i)15-s + (−0.5 − 0.866i)16-s + (−2.86 − 2.96i)17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.866 − 0.499i)3-s + (0.249 − 0.433i)4-s + (0.327 + 1.22i)5-s + (0.353 − 0.612i)6-s + (−0.101 + 0.377i)7-s − 0.353i·8-s + (0.5 − 0.866i)9-s + (0.632 + 0.632i)10-s + (0.180 − 0.672i)11-s − 0.499i·12-s + (−0.480 + 0.832i)13-s + (0.0716 + 0.267i)14-s + (0.894 + 0.894i)15-s + (−0.125 − 0.216i)16-s + (−0.695 − 0.718i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 306 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.883 + 0.469i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 306 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.883 + 0.469i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(306\)    =    \(2 \cdot 3^{2} \cdot 17\)
Sign: $0.883 + 0.469i$
Analytic conductor: \(2.44342\)
Root analytic conductor: \(1.56314\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{306} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 306,\ (\ :1/2),\ 0.883 + 0.469i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.28461 - 0.569235i\)
\(L(\frac12)\) \(\approx\) \(2.28461 - 0.569235i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 + (-1.5 + 0.866i)T \)
17 \( 1 + (2.86 + 2.96i)T \)
good5 \( 1 + (-0.732 - 2.73i)T + (-4.33 + 2.5i)T^{2} \)
7 \( 1 + (0.267 - i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (-0.598 + 2.23i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (1.73 - 3i)T + (-6.5 - 11.2i)T^{2} \)
19 \( 1 - iT - 19T^{2} \)
23 \( 1 + (1.73 - 0.464i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (8.46 + 2.26i)T + (25.1 + 14.5i)T^{2} \)
31 \( 1 + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (-3.46 + 3.46i)T - 37iT^{2} \)
41 \( 1 + (9.96 - 2.66i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-4.5 + 2.59i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5 - 8.66i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 2.53iT - 53T^{2} \)
59 \( 1 + (1.5 + 0.866i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.73 - 10.1i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (-2.33 + 4.03i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-5.46 + 5.46i)T - 71iT^{2} \)
73 \( 1 + (-3.90 + 3.90i)T - 73iT^{2} \)
79 \( 1 + (3.26 - 12.1i)T + (-68.4 - 39.5i)T^{2} \)
83 \( 1 + (3.92 - 2.26i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 6.92T + 89T^{2} \)
97 \( 1 + (-2.5 - 0.669i)T + (84.0 + 48.5i)T^{2} \)
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

−11.67980789504321069106465139301, −10.89677013057487090164591835812, −9.707577654892109351613348200498, −9.003967663767065765671817277976, −7.55219792334173888767920708338, −6.73416480458243173783576598281, −5.86117917625732302356569576868, −4.10093325607851983085446613596, −2.94562431232043103001305459047, −2.09886256556674953146134406542, 2.03728538653477675078227100550, 3.68365248292504394405724867058, 4.64616266967937049677917706046, 5.47332780945439058344234039649, 7.02626964578447871367675661289, 8.042324171369145027810708743588, 8.877955146193325281976418936679, 9.753320357933702752749755033599, 10.70620707526069557131684590472, 12.13492565688694710290045183190

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