↑

Properties

Label 2-720-16.13-c1-0-23
Degree $2$
Conductor $720$
Sign $0.856 + 0.515i$
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  + (1.32 − 0.492i)2-s + (1.51 − 1.30i)4-s + (−0.707 + 0.707i)5-s + 2.05i·7-s + (1.36 − 2.47i)8-s + (−0.589 + 1.28i)10-s + (2.89 − 2.89i)11-s + (−0.887 − 0.887i)13-s + (1.01 + 2.72i)14-s + (0.592 − 3.95i)16-s + 7.70·17-s + (1.96 + 1.96i)19-s + (−0.148 + 1.99i)20-s + (2.41 − 5.26i)22-s − 1.75i·23-s + ⋯
L(s)  = 1  + (0.937 − 0.348i)2-s + (0.757 − 0.652i)4-s + (−0.316 + 0.316i)5-s + 0.776i·7-s + (0.483 − 0.875i)8-s + (−0.186 + 0.406i)10-s + (0.873 − 0.873i)11-s + (−0.246 − 0.246i)13-s + (0.270 + 0.727i)14-s + (0.148 − 0.988i)16-s + 1.86·17-s + (0.450 + 0.450i)19-s + (−0.0332 + 0.445i)20-s + (0.515 − 1.12i)22-s − 0.365i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.856 + 0.515i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.856 + 0.515i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $0.856 + 0.515i$
Analytic conductor: \(5.74922\)
Root analytic conductor: \(2.39775\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (541, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :1/2),\ 0.856 + 0.515i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.65207 - 0.736095i\)
\(L(\frac12)\) \(\approx\) \(2.65207 - 0.736095i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.32 + 0.492i)T \)
3 \( 1 \)
5 \( 1 + (0.707 - 0.707i)T \)
good7 \( 1 - 2.05iT - 7T^{2} \)
11 \( 1 + (-2.89 + 2.89i)T - 11iT^{2} \)
13 \( 1 + (0.887 + 0.887i)T + 13iT^{2} \)
17 \( 1 - 7.70T + 17T^{2} \)
19 \( 1 + (-1.96 - 1.96i)T + 19iT^{2} \)
23 \( 1 + 1.75iT - 23T^{2} \)
29 \( 1 + (-1.03 - 1.03i)T + 29iT^{2} \)
31 \( 1 - 1.03T + 31T^{2} \)
37 \( 1 + (7.76 - 7.76i)T - 37iT^{2} \)
41 \( 1 + 1.08iT - 41T^{2} \)
43 \( 1 + (4.29 - 4.29i)T - 43iT^{2} \)
47 \( 1 + 8.19T + 47T^{2} \)
53 \( 1 + (-3.76 + 3.76i)T - 53iT^{2} \)
59 \( 1 + (3.92 - 3.92i)T - 59iT^{2} \)
61 \( 1 + (6.18 + 6.18i)T + 61iT^{2} \)
67 \( 1 + (8.26 + 8.26i)T + 67iT^{2} \)
71 \( 1 + 6.34iT - 71T^{2} \)
73 \( 1 - 14.3iT - 73T^{2} \)
79 \( 1 + 16.1T + 79T^{2} \)
83 \( 1 + (2.72 + 2.72i)T + 83iT^{2} \)
89 \( 1 - 7.96iT - 89T^{2} \)
97 \( 1 - 7.11T + 97T^{2} \)
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

−10.41521853896663811256088876979, −9.769364199201057092682771900706, −8.585061675616380137212431953542, −7.62559941789488941869801928000, −6.52013744641109656308631256442, −5.77785985327410311719678205562, −4.92784256617939886314320808696, −3.53973888398458752317144877246, −3.01076176237371225876235977317, −1.38107446223686706897142656201, 1.53147397433865600183037867026, 3.23994398858375432615666458560, 4.09001784943550083066879856404, 4.94520856534559687776011618703, 5.92645971455613676182867016890, 7.28388862259853371473222399095, 7.31628359050260231245088997189, 8.597218688813319926296260279692, 9.722289652615740093730952571791, 10.54604545463746698533780081545

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