↑

Properties

Label 2-360-9.7-c1-0-9
Degree $2$
Conductor $360$
Sign $0.308 + 0.951i$
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  + (1.24 − 1.20i)3-s + (−0.5 − 0.866i)5-s + (−0.433 + 0.751i)7-s + (0.106 − 2.99i)9-s + (2.04 − 3.53i)11-s + (−0.606 − 1.05i)13-s + (−1.66 − 0.477i)15-s + 7.82·17-s − 4.51·19-s + (0.363 + 1.45i)21-s + (−1.43 − 2.48i)23-s + (−0.499 + 0.866i)25-s + (−3.47 − 3.86i)27-s + (−3.14 + 5.45i)29-s + (1.26 + 2.18i)31-s + ⋯
L(s)  = 1  + (0.719 − 0.694i)3-s + (−0.223 − 0.387i)5-s + (−0.163 + 0.284i)7-s + (0.0354 − 0.999i)9-s + (0.616 − 1.06i)11-s + (−0.168 − 0.291i)13-s + (−0.429 − 0.123i)15-s + 1.89·17-s − 1.03·19-s + (0.0792 + 0.318i)21-s + (−0.298 − 0.517i)23-s + (−0.0999 + 0.173i)25-s + (−0.668 − 0.743i)27-s + (−0.584 + 1.01i)29-s + (0.226 + 0.392i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.308 + 0.951i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.308 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $0.308 + 0.951i$
Analytic conductor: \(2.87461\)
Root analytic conductor: \(1.69546\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{360} (241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 360,\ (\ :1/2),\ 0.308 + 0.951i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.30277 - 0.947028i\)
\(L(\frac12)\) \(\approx\) \(1.30277 - 0.947028i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-1.24 + 1.20i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
good7 \( 1 + (0.433 - 0.751i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.04 + 3.53i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.606 + 1.05i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 7.82T + 17T^{2} \)
19 \( 1 + 4.51T + 19T^{2} \)
23 \( 1 + (1.43 + 2.48i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.14 - 5.45i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.26 - 2.18i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 0.523T + 37T^{2} \)
41 \( 1 + (-4.06 - 7.03i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.91 - 3.31i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.695 + 1.20i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 13.9T + 53T^{2} \)
59 \( 1 + (3.43 + 5.95i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.54 - 7.86i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.68 + 2.92i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.21T + 71T^{2} \)
73 \( 1 - 8.60T + 73T^{2} \)
79 \( 1 + (8.12 - 14.0i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.22 - 5.57i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 10.2T + 89T^{2} \)
97 \( 1 + (-4.38 + 7.59i)T + (-48.5 - 84.0i)T^{2} \)
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

−11.47581956356219672442941205013, −10.23290956841334372127946729208, −9.152569646624355457692287887283, −8.437070003589888625038082367968, −7.68237016315858686794114402372, −6.49274855052734459468186354327, −5.55969320833444285473176723918, −3.89209655455574016593125830714, −2.87990850527980457662160071803, −1.15660153022412264311634923054, 2.12811568987222358514484175324, 3.59426149047292166809137467688, 4.33300827274907977806426228187, 5.71394856308343010638094411932, 7.14760624198391513180654551173, 7.82204894085814226240835392709, 8.989766312748683521838054089460, 9.911045516001513323156127436979, 10.36576445293342859090620411448, 11.61296841056384630621173132821

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