↑

Properties

Label 2-360-72.11-c1-0-29
Degree $2$
Conductor $360$
Sign $-0.270 + 0.962i$
Analytic cond. $2.87461$
Root an. cond. $1.69546$
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  + (−0.877 − 1.10i)2-s + (0.875 − 1.49i)3-s + (−0.461 + 1.94i)4-s + (0.5 + 0.866i)5-s + (−2.42 + 0.339i)6-s + (1.05 + 0.608i)7-s + (2.56 − 1.19i)8-s + (−1.46 − 2.61i)9-s + (0.522 − 1.31i)10-s + (−2.23 − 1.29i)11-s + (2.50 + 2.39i)12-s + (4.16 − 2.40i)13-s + (−0.249 − 1.70i)14-s + (1.73 + 0.0107i)15-s + (−3.57 − 1.79i)16-s − 5.97i·17-s + ⋯
L(s)  = 1  + (−0.620 − 0.784i)2-s + (0.505 − 0.862i)3-s + (−0.230 + 0.973i)4-s + (0.223 + 0.387i)5-s + (−0.990 + 0.138i)6-s + (0.398 + 0.229i)7-s + (0.906 − 0.422i)8-s + (−0.489 − 0.872i)9-s + (0.165 − 0.415i)10-s + (−0.673 − 0.389i)11-s + (0.723 + 0.690i)12-s + (1.15 − 0.667i)13-s + (−0.0666 − 0.454i)14-s + (0.447 + 0.00278i)15-s + (−0.893 − 0.448i)16-s − 1.44i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.270 + 0.962i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.270 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $-0.270 + 0.962i$
Analytic conductor: \(2.87461\)
Root analytic conductor: \(1.69546\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{360} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 360,\ (\ :1/2),\ -0.270 + 0.962i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.726995 - 0.959887i\)
\(L(\frac12)\) \(\approx\) \(0.726995 - 0.959887i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.877 + 1.10i)T \)
3 \( 1 + (-0.875 + 1.49i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
good7 \( 1 + (-1.05 - 0.608i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.23 + 1.29i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-4.16 + 2.40i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 5.97iT - 17T^{2} \)
19 \( 1 - 4.69T + 19T^{2} \)
23 \( 1 + (-0.866 - 1.50i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.85 + 6.67i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.30 - 2.48i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 8.33iT - 37T^{2} \)
41 \( 1 + (2.00 - 1.16i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.97 - 3.41i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.25 - 5.63i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 0.0729T + 53T^{2} \)
59 \( 1 + (-5.32 + 3.07i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-9.89 - 5.71i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.59 - 4.49i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 9.50T + 71T^{2} \)
73 \( 1 - 10.7T + 73T^{2} \)
79 \( 1 + (9.17 + 5.29i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.02 - 2.90i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 11.5iT - 89T^{2} \)
97 \( 1 + (-4.84 + 8.39i)T + (-48.5 - 84.0i)T^{2} \)
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

−11.44010289479479909406303025965, −10.21564684172023435076050299819, −9.316568199476522643306416799060, −8.310408286575660624901193306499, −7.75299347781774343505311905306, −6.67207113876693064052435278977, −5.27737468660590670127696858820, −3.37751938325814283402639569626, −2.61756371050031615179108619926, −1.08670017935514459509074164117, 1.76660478516435085151102378573, 3.82564218875992632805638541331, 4.94855418273905003181862226384, 5.81074077130606362503975212636, 7.17373445628042018987146255141, 8.275950240021628446274489625705, 8.767600517698696951201429576094, 9.729439132476472168379499495381, 10.56436783803326889035798858450, 11.19786561652176421623821907493

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