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Properties

Label 2-120-24.11-c3-0-0
Degree $2$
Conductor $120$
Sign $0.654 - 0.755i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.638 − 2.75i)2-s + (−0.899 − 5.11i)3-s + (−7.18 + 3.51i)4-s − 5·5-s + (−13.5 + 5.74i)6-s + 23.1i·7-s + (14.2 + 17.5i)8-s + (−25.3 + 9.21i)9-s + (3.19 + 13.7i)10-s − 5.98i·11-s + (24.4 + 33.6i)12-s − 3.38i·13-s + (63.7 − 14.7i)14-s + (4.49 + 25.5i)15-s + (39.2 − 50.5i)16-s + 35.8i·17-s + ⋯
L(s)  = 1  + (−0.225 − 0.974i)2-s + (−0.173 − 0.984i)3-s + (−0.898 + 0.439i)4-s − 0.447·5-s + (−0.920 + 0.391i)6-s + 1.24i·7-s + (0.631 + 0.775i)8-s + (−0.940 + 0.341i)9-s + (0.100 + 0.435i)10-s − 0.163i·11-s + (0.588 + 0.808i)12-s − 0.0722i·13-s + (1.21 − 0.281i)14-s + (0.0774 + 0.440i)15-s + (0.613 − 0.789i)16-s + 0.512i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.654 - 0.755i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.654 - 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(120\)    =    \(2^{3} \cdot 3 \cdot 5\)
Sign: $0.654 - 0.755i$
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: \((2,\ 120,\ (\ :3/2),\ 0.654 - 0.755i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.364914 + 0.166718i\)
\(L(\frac12)\) \(\approx\) \(0.364914 + 0.166718i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.638 + 2.75i)T \)
3 \( 1 + (0.899 + 5.11i)T \)
5 \( 1 + 5T \)
good7 \( 1 - 23.1iT - 343T^{2} \)
11 \( 1 + 5.98iT - 1.33e3T^{2} \)
13 \( 1 + 3.38iT - 2.19e3T^{2} \)
17 \( 1 - 35.8iT - 4.91e3T^{2} \)
19 \( 1 + 51.7T + 6.85e3T^{2} \)
23 \( 1 + 123.T + 1.21e4T^{2} \)
29 \( 1 - 130.T + 2.43e4T^{2} \)
31 \( 1 - 298. iT - 2.97e4T^{2} \)
37 \( 1 - 343. iT - 5.06e4T^{2} \)
41 \( 1 + 121. iT - 6.89e4T^{2} \)
43 \( 1 + 535.T + 7.95e4T^{2} \)
47 \( 1 + 401.T + 1.03e5T^{2} \)
53 \( 1 + 67.6T + 1.48e5T^{2} \)
59 \( 1 + 717. iT - 2.05e5T^{2} \)
61 \( 1 - 45.2iT - 2.26e5T^{2} \)
67 \( 1 - 722.T + 3.00e5T^{2} \)
71 \( 1 + 1.03e3T + 3.57e5T^{2} \)
73 \( 1 + 331.T + 3.89e5T^{2} \)
79 \( 1 + 449. iT - 4.93e5T^{2} \)
83 \( 1 - 641. iT - 5.71e5T^{2} \)
89 \( 1 - 855. iT - 7.04e5T^{2} \)
97 \( 1 - 82.0T + 9.12e5T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.72161411473512450982703828428, −12.10994804326994203390756875317, −11.41042090561628255953548701676, −10.16847871659690084731953921323, −8.603411213062947560565449954714, −8.206175656080597685276700461328, −6.47951688979416837525345644409, −5.05726707851233587016320199776, −3.12721360812574683949648858289, −1.75742295510809084179662393520, 0.23409160392708866879567582762, 3.87123734851352623304699658606, 4.67399693536398416312666843079, 6.16020662029145614152884881319, 7.41487675472495917025412517650, 8.463327166336749535279187604844, 9.740172428557143281669417442496, 10.42877554485114372720842390361, 11.60034420564073458291804733020, 13.20548837750953803840933670977

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