↑

Properties

Label 2-120-24.11-c3-0-26
Degree $2$
Conductor $120$
Sign $0.336 - 0.941i$
Analytic cond. $7.08022$
Root an. cond. $2.66087$
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  + (2.71 + 0.804i)2-s + (2.58 + 4.50i)3-s + (6.70 + 4.36i)4-s + 5·5-s + (3.39 + 14.2i)6-s − 6.99i·7-s + (14.6 + 17.2i)8-s + (−13.5 + 23.3i)9-s + (13.5 + 4.02i)10-s − 7.63i·11-s + (−2.29 + 41.5i)12-s − 45.0i·13-s + (5.62 − 18.9i)14-s + (12.9 + 22.5i)15-s + (25.9 + 58.5i)16-s + 7.59i·17-s + ⋯
L(s)  = 1  + (0.958 + 0.284i)2-s + (0.498 + 0.867i)3-s + (0.838 + 0.545i)4-s + 0.447·5-s + (0.231 + 0.972i)6-s − 0.377i·7-s + (0.648 + 0.761i)8-s + (−0.503 + 0.864i)9-s + (0.428 + 0.127i)10-s − 0.209i·11-s + (−0.0552 + 0.998i)12-s − 0.962i·13-s + (0.107 − 0.361i)14-s + (0.222 + 0.387i)15-s + (0.405 + 0.914i)16-s + 0.108i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.336 - 0.941i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.336 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(120\)    =    \(2^{3} \cdot 3 \cdot 5\)
Sign: $0.336 - 0.941i$
Analytic conductor: \(7.08022\)
Root analytic conductor: \(2.66087\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{120} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 120,\ (\ :3/2),\ 0.336 - 0.941i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.77706 + 1.95564i\)
\(L(\frac12)\) \(\approx\) \(2.77706 + 1.95564i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.71 - 0.804i)T \)
3 \( 1 + (-2.58 - 4.50i)T \)
5 \( 1 - 5T \)
good7 \( 1 + 6.99iT - 343T^{2} \)
11 \( 1 + 7.63iT - 1.33e3T^{2} \)
13 \( 1 + 45.0iT - 2.19e3T^{2} \)
17 \( 1 - 7.59iT - 4.91e3T^{2} \)
19 \( 1 + 125.T + 6.85e3T^{2} \)
23 \( 1 - 57.3T + 1.21e4T^{2} \)
29 \( 1 - 190.T + 2.43e4T^{2} \)
31 \( 1 + 96.2iT - 2.97e4T^{2} \)
37 \( 1 + 174. iT - 5.06e4T^{2} \)
41 \( 1 + 353. iT - 6.89e4T^{2} \)
43 \( 1 - 96.3T + 7.95e4T^{2} \)
47 \( 1 + 120.T + 1.03e5T^{2} \)
53 \( 1 + 677.T + 1.48e5T^{2} \)
59 \( 1 - 767. iT - 2.05e5T^{2} \)
61 \( 1 + 605. iT - 2.26e5T^{2} \)
67 \( 1 + 464.T + 3.00e5T^{2} \)
71 \( 1 + 110.T + 3.57e5T^{2} \)
73 \( 1 - 716.T + 3.89e5T^{2} \)
79 \( 1 + 646. iT - 4.93e5T^{2} \)
83 \( 1 - 1.26e3iT - 5.71e5T^{2} \)
89 \( 1 + 302. iT - 7.04e5T^{2} \)
97 \( 1 - 160.T + 9.12e5T^{2} \)
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

−13.41027159528115137243076933203, −12.49085108815312952461966265752, −10.92824959980237663032838305526, −10.36861387796814729272476658832, −8.829379872325057804562590822713, −7.75080171323203507578126681453, −6.23691046697560997471062270208, −5.04514721396351029855087335573, −3.87202314022709063403264944834, −2.53635277634837197997607052799, 1.66317290471445233752978972466, 2.86248883544642159294474498931, 4.58527362169684932362620565289, 6.18541396441541410020640037532, 6.88281531291862944088895554335, 8.431084114025765561080912040419, 9.663368235868695728079408932950, 11.04927603036402111220920328814, 12.12760058760638910831735994890, 12.83990668813581160806809846788

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