↑

Properties

Label 2-720-16.5-c1-0-35
Degree $2$
Conductor $720$
Sign $-0.944 - 0.329i$
Analytic cond. $5.74922$
Root an. cond. $2.39775$
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  + (−1.18 − 0.768i)2-s + (0.817 + 1.82i)4-s + (−0.707 − 0.707i)5-s − 4.92i·7-s + (0.432 − 2.79i)8-s + (0.295 + 1.38i)10-s + (−2.45 − 2.45i)11-s + (−2.93 + 2.93i)13-s + (−3.78 + 5.84i)14-s + (−2.66 + 2.98i)16-s + 5.77·17-s + (−0.984 + 0.984i)19-s + (0.712 − 1.86i)20-s + (1.02 + 4.80i)22-s + 0.539i·23-s + ⋯
L(s)  = 1  + (−0.839 − 0.543i)2-s + (0.408 + 0.912i)4-s + (−0.316 − 0.316i)5-s − 1.86i·7-s + (0.152 − 0.988i)8-s + (0.0935 + 0.437i)10-s + (−0.741 − 0.741i)11-s + (−0.812 + 0.812i)13-s + (−1.01 + 1.56i)14-s + (−0.665 + 0.746i)16-s + 1.40·17-s + (−0.225 + 0.225i)19-s + (0.159 − 0.417i)20-s + (0.219 + 1.02i)22-s + 0.112i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.944 - 0.329i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.944 - 0.329i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $-0.944 - 0.329i$
Analytic conductor: \(5.74922\)
Root analytic conductor: \(2.39775\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :1/2),\ -0.944 - 0.329i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0640500 + 0.378329i\)
\(L(\frac12)\) \(\approx\) \(0.0640500 + 0.378329i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.18 + 0.768i)T \)
3 \( 1 \)
5 \( 1 + (0.707 + 0.707i)T \)
good7 \( 1 + 4.92iT - 7T^{2} \)
11 \( 1 + (2.45 + 2.45i)T + 11iT^{2} \)
13 \( 1 + (2.93 - 2.93i)T - 13iT^{2} \)
17 \( 1 - 5.77T + 17T^{2} \)
19 \( 1 + (0.984 - 0.984i)T - 19iT^{2} \)
23 \( 1 - 0.539iT - 23T^{2} \)
29 \( 1 + (6.81 - 6.81i)T - 29iT^{2} \)
31 \( 1 + 2.63T + 31T^{2} \)
37 \( 1 + (6.00 + 6.00i)T + 37iT^{2} \)
41 \( 1 + 5.17iT - 41T^{2} \)
43 \( 1 + (0.180 + 0.180i)T + 43iT^{2} \)
47 \( 1 + 5.57T + 47T^{2} \)
53 \( 1 + (-0.146 - 0.146i)T + 53iT^{2} \)
59 \( 1 + (3.13 + 3.13i)T + 59iT^{2} \)
61 \( 1 + (1.87 - 1.87i)T - 61iT^{2} \)
67 \( 1 + (-8.02 + 8.02i)T - 67iT^{2} \)
71 \( 1 - 7.40iT - 71T^{2} \)
73 \( 1 - 11.1iT - 73T^{2} \)
79 \( 1 - 7.71T + 79T^{2} \)
83 \( 1 + (-1.62 + 1.62i)T - 83iT^{2} \)
89 \( 1 + 9.54iT - 89T^{2} \)
97 \( 1 - 4.39T + 97T^{2} \)
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

−10.04225839007141845755317926721, −9.225654541766467562991243421945, −8.136426300983730370878940884117, −7.46306911119683088835448264777, −6.92968181658531364620581921765, −5.28894421092249697669002318040, −3.99739652721955708900774583342, −3.33547021714658020860083360346, −1.59615042558229627364336348413, −0.25508049756672141514650214960, 2.06391524688856767100796786526, 2.97002749358258667132408037101, 5.06109254434597633975538561549, 5.54447651757388392355813224949, 6.55694268333125851036157535083, 7.81398984564937529256640643248, 8.016525980767338433166211544440, 9.249445243180421604871860171848, 9.808793818832075740018317121151, 10.62745356971452614072244332616

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