↑

Properties

Label 2-720-5.4-c3-0-2
Degree $2$
Conductor $720$
Sign $-0.995 - 0.0894i$
Analytic cond. $42.4813$
Root an. cond. $6.51777$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−11.1 − i)5-s + 22.2i·7-s + 22.2·11-s − 66.8i·13-s + 62i·17-s + 84·19-s + 140i·23-s + (122. + 22.2i)25-s − 200.·29-s − 16·31-s + (22.2 − 247. i)35-s + 244. i·37-s − 222.·41-s − 356. i·43-s − 100i·47-s + ⋯
L(s)  = 1  + (−0.995 − 0.0894i)5-s + 1.20i·7-s + 0.610·11-s − 1.42i·13-s + 0.884i·17-s + 1.01·19-s + 1.26i·23-s + (0.983 + 0.178i)25-s − 1.28·29-s − 0.0926·31-s + (0.107 − 1.19i)35-s + 1.08i·37-s − 0.848·41-s − 1.26i·43-s − 0.310i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.995 - 0.0894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $-0.995 - 0.0894i$
Analytic conductor: \(42.4813\)
Root analytic conductor: \(6.51777\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :3/2),\ -0.995 - 0.0894i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.3819371610\)
\(L(\frac12)\) \(\approx\) \(0.3819371610\)
\(L(\frac{5}{2})\) 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 \( 1 \)
5 \( 1 + (11.1 + i)T \)
good7 \( 1 - 22.2iT - 343T^{2} \)
11 \( 1 - 22.2T + 1.33e3T^{2} \)
13 \( 1 + 66.8iT - 2.19e3T^{2} \)
17 \( 1 - 62iT - 4.91e3T^{2} \)
19 \( 1 - 84T + 6.85e3T^{2} \)
23 \( 1 - 140iT - 1.21e4T^{2} \)
29 \( 1 + 200.T + 2.43e4T^{2} \)
31 \( 1 + 16T + 2.97e4T^{2} \)
37 \( 1 - 244. iT - 5.06e4T^{2} \)
41 \( 1 + 222.T + 6.89e4T^{2} \)
43 \( 1 + 356. iT - 7.95e4T^{2} \)
47 \( 1 + 100iT - 1.03e5T^{2} \)
53 \( 1 + 738iT - 1.48e5T^{2} \)
59 \( 1 + 645.T + 2.05e5T^{2} \)
61 \( 1 + 358T + 2.26e5T^{2} \)
67 \( 1 - 846. iT - 3.00e5T^{2} \)
71 \( 1 + 935.T + 3.57e5T^{2} \)
73 \( 1 + 445. iT - 3.89e5T^{2} \)
79 \( 1 + 936T + 4.93e5T^{2} \)
83 \( 1 - 1.30e3iT - 5.71e5T^{2} \)
89 \( 1 + 712.T + 7.04e5T^{2} \)
97 \( 1 - 757. iT - 9.12e5T^{2} \)
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

−10.45398525440194721541906244553, −9.471788631953932144745848309202, −8.611130596880394879829715338918, −7.952273429759176090975416010673, −7.07766964346016284866986277612, −5.78393044010727112382474492281, −5.20651583652282884583159054747, −3.77134265804284631541550863091, −3.05261430287834379294886120602, −1.47551236156233028737003320987, 0.11337473905360463162068022596, 1.36824320924121052022578301354, 3.08938486834435412956095564357, 4.14675451323906191403912168143, 4.63561666947938479953461216749, 6.24411014741662030835948863896, 7.27855786533351750121109859551, 7.48071553794630749255454729862, 8.847012839741067404553420373598, 9.489741943841659917766000385339

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