↑

Properties

Label 2-120-120.59-c3-0-39
Degree $2$
Conductor $120$
Sign $0.274 + 0.961i$
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  + (−2.22 − 1.74i)2-s + (4.99 − 1.43i)3-s + (1.93 + 7.76i)4-s + (−10.2 + 4.55i)5-s + (−13.6 − 5.51i)6-s + 3.90·7-s + (9.22 − 20.6i)8-s + (22.9 − 14.2i)9-s + (30.6 + 7.64i)10-s − 59.1i·11-s + (20.7 + 36.0i)12-s + 63.6·13-s + (−8.70 − 6.80i)14-s + (−44.5 + 37.3i)15-s + (−56.5 + 29.9i)16-s + 69.6·17-s + ⋯
L(s)  = 1  + (−0.787 − 0.615i)2-s + (0.961 − 0.275i)3-s + (0.241 + 0.970i)4-s + (−0.913 + 0.407i)5-s + (−0.926 − 0.375i)6-s + 0.210·7-s + (0.407 − 0.913i)8-s + (0.848 − 0.529i)9-s + (0.970 + 0.241i)10-s − 1.62i·11-s + (0.499 + 0.866i)12-s + 1.35·13-s + (−0.166 − 0.129i)14-s + (−0.765 + 0.642i)15-s + (−0.883 + 0.468i)16-s + 0.993·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.274 + 0.961i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.274 + 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(120\)    =    \(2^{3} \cdot 3 \cdot 5\)
Sign: $0.274 + 0.961i$
Analytic conductor: \(7.08022\)
Root analytic conductor: \(2.66087\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{120} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 120,\ (\ :3/2),\ 0.274 + 0.961i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.11611 - 0.841709i\)
\(L(\frac12)\) \(\approx\) \(1.11611 - 0.841709i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.22 + 1.74i)T \)
3 \( 1 + (-4.99 + 1.43i)T \)
5 \( 1 + (10.2 - 4.55i)T \)
good7 \( 1 - 3.90T + 343T^{2} \)
11 \( 1 + 59.1iT - 1.33e3T^{2} \)
13 \( 1 - 63.6T + 2.19e3T^{2} \)
17 \( 1 - 69.6T + 4.91e3T^{2} \)
19 \( 1 - 33.0T + 6.85e3T^{2} \)
23 \( 1 + 90.6iT - 1.21e4T^{2} \)
29 \( 1 + 172.T + 2.43e4T^{2} \)
31 \( 1 + 61.7iT - 2.97e4T^{2} \)
37 \( 1 - 10.7T + 5.06e4T^{2} \)
41 \( 1 - 475. iT - 6.89e4T^{2} \)
43 \( 1 + 59.3iT - 7.95e4T^{2} \)
47 \( 1 - 500. iT - 1.03e5T^{2} \)
53 \( 1 + 407. iT - 1.48e5T^{2} \)
59 \( 1 + 17.5iT - 2.05e5T^{2} \)
61 \( 1 + 245. iT - 2.26e5T^{2} \)
67 \( 1 + 35.7iT - 3.00e5T^{2} \)
71 \( 1 - 889.T + 3.57e5T^{2} \)
73 \( 1 - 617. iT - 3.89e5T^{2} \)
79 \( 1 + 108. iT - 4.93e5T^{2} \)
83 \( 1 - 628.T + 5.71e5T^{2} \)
89 \( 1 - 763. iT - 7.04e5T^{2} \)
97 \( 1 - 866. iT - 9.12e5T^{2} \)
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

−12.71679568278368720517651578166, −11.47467633593362322027855479364, −10.85585771542719559912010718294, −9.455992026320920378205026907691, −8.252175287434550092908911758023, −7.982991884679194434209737880894, −6.48804389793244257029926575495, −3.79506748721267942519844160594, −3.05070279986755862695519478143, −1.01200754009842438148616136250, 1.57106548915839942776122218170, 3.80572968340200023170056499490, 5.20489421087130044160239080381, 7.18181312590591652066989425360, 7.83572260521266257542564960635, 8.841816852474679430827210854380, 9.715842443971129840650637546424, 10.83413012643085445710599155825, 12.13481544873146718119826027817, 13.44497706955114544141324777027

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