↑

Properties

Label 2-720-180.139-c0-0-1
Degree $2$
Conductor $720$
Sign $0.939 + 0.342i$
Analytic cond. $0.359326$
Root an. cond. $0.599438$
Motivic weight $0$
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)3-s + (0.5 − 0.866i)5-s + (−0.866 − 1.5i)7-s + (0.499 + 0.866i)9-s + (0.866 − 0.499i)15-s − 1.73i·21-s + (−0.866 + 1.5i)23-s + (−0.499 − 0.866i)25-s + 0.999i·27-s + (0.5 + 0.866i)29-s − 1.73·35-s + (−0.5 + 0.866i)41-s + 0.999·45-s + (0.866 + 1.5i)47-s + (−1 + 1.73i)49-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)3-s + (0.5 − 0.866i)5-s + (−0.866 − 1.5i)7-s + (0.499 + 0.866i)9-s + (0.866 − 0.499i)15-s − 1.73i·21-s + (−0.866 + 1.5i)23-s + (−0.499 − 0.866i)25-s + 0.999i·27-s + (0.5 + 0.866i)29-s − 1.73·35-s + (−0.5 + 0.866i)41-s + 0.999·45-s + (0.866 + 1.5i)47-s + (−1 + 1.73i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $0.939 + 0.342i$
Analytic conductor: \(0.359326\)
Root analytic conductor: \(0.599438\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (319, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :0),\ 0.939 + 0.342i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.234830524\)
\(L(\frac12)\) \(\approx\) \(1.234830524\)
\(L(1)\) 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 \( 1 + (-0.866 - 0.5i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
good7 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 + T + T^{2} \)
97 \( 1 + (0.5 - 0.866i)T^{2} \)
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

−10.20029961747774673707224432113, −9.755916473043659594658264836448, −9.048397663310796971925852685668, −8.010195232001580241879209933299, −7.28891948476599430707801398791, −6.14834496880798090102101709635, −4.89485694381613977259459875115, −4.03758408865111910624035305607, −3.15618045760290732572839988046, −1.51478032317578336594166757865, 2.24611297220220331476510645754, 2.70293794684888481033897050418, 3.87276349712210614788700350242, 5.61399028497859415870112415434, 6.36866776584097815883184440209, 7.02585684708879847186149528269, 8.270797762792726198707625039532, 8.892380100205280971221178674388, 9.766825487891479562477886837017, 10.35522067596290242243292923170

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