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Properties

Label 2-120-120.53-c3-0-31
Degree $2$
Conductor $120$
Sign $0.856 - 0.515i$
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  + (2.75 + 0.631i)2-s + (−4.07 − 3.22i)3-s + (7.20 + 3.48i)4-s + (8.53 + 7.21i)5-s + (−9.20 − 11.4i)6-s + (−9.44 + 9.44i)7-s + (17.6 + 14.1i)8-s + (6.24 + 26.2i)9-s + (18.9 + 25.2i)10-s + 41.6·11-s + (−18.1 − 37.4i)12-s + (53.3 − 53.3i)13-s + (−32.0 + 20.0i)14-s + (−11.5 − 56.9i)15-s + (39.7 + 50.1i)16-s + (41.4 + 41.4i)17-s + ⋯
L(s)  = 1  + (0.974 + 0.223i)2-s + (−0.784 − 0.620i)3-s + (0.900 + 0.435i)4-s + (0.763 + 0.645i)5-s + (−0.626 − 0.779i)6-s + (−0.510 + 0.510i)7-s + (0.780 + 0.625i)8-s + (0.231 + 0.972i)9-s + (0.600 + 0.799i)10-s + 1.14·11-s + (−0.436 − 0.899i)12-s + (1.13 − 1.13i)13-s + (−0.611 + 0.383i)14-s + (−0.198 − 0.980i)15-s + (0.620 + 0.784i)16-s + (0.591 + 0.591i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.856 - 0.515i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.856 - 0.515i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(120\)    =    \(2^{3} \cdot 3 \cdot 5\)
Sign: $0.856 - 0.515i$
Analytic conductor: \(7.08022\)
Root analytic conductor: \(2.66087\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{120} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 120,\ (\ :3/2),\ 0.856 - 0.515i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.51219 + 0.697319i\)
\(L(\frac12)\) \(\approx\) \(2.51219 + 0.697319i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.75 - 0.631i)T \)
3 \( 1 + (4.07 + 3.22i)T \)
5 \( 1 + (-8.53 - 7.21i)T \)
good7 \( 1 + (9.44 - 9.44i)T - 343iT^{2} \)
11 \( 1 - 41.6T + 1.33e3T^{2} \)
13 \( 1 + (-53.3 + 53.3i)T - 2.19e3iT^{2} \)
17 \( 1 + (-41.4 - 41.4i)T + 4.91e3iT^{2} \)
19 \( 1 + 145.T + 6.85e3T^{2} \)
23 \( 1 + (-27.2 + 27.2i)T - 1.21e4iT^{2} \)
29 \( 1 - 107. iT - 2.43e4T^{2} \)
31 \( 1 + 151.T + 2.97e4T^{2} \)
37 \( 1 + (284. + 284. i)T + 5.06e4iT^{2} \)
41 \( 1 + 203. iT - 6.89e4T^{2} \)
43 \( 1 + (-194. + 194. i)T - 7.95e4iT^{2} \)
47 \( 1 + (197. + 197. i)T + 1.03e5iT^{2} \)
53 \( 1 + (202. + 202. i)T + 1.48e5iT^{2} \)
59 \( 1 + 460. iT - 2.05e5T^{2} \)
61 \( 1 - 94.5iT - 2.26e5T^{2} \)
67 \( 1 + (-258. - 258. i)T + 3.00e5iT^{2} \)
71 \( 1 + 76.8iT - 3.57e5T^{2} \)
73 \( 1 + (-14.0 - 14.0i)T + 3.89e5iT^{2} \)
79 \( 1 - 221. iT - 4.93e5T^{2} \)
83 \( 1 + (38.4 + 38.4i)T + 5.71e5iT^{2} \)
89 \( 1 + 230.T + 7.04e5T^{2} \)
97 \( 1 + (-758. + 758. i)T - 9.12e5iT^{2} \)
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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

−12.83497954889581214214004164013, −12.52890147747600200333807332008, −11.06998396097544548602854243494, −10.49991148461570502667701101434, −8.600956317551280758748169977073, −7.00698589514958921732367987330, −6.18801212394933060224684548746, −5.56085316854767039780684320393, −3.59083252447403318562939412125, −1.89143592833242456771719557399, 1.36610216087989832299820429916, 3.77036271635818135562122281891, 4.67755381829312107188720441209, 6.12817145255476764408466081484, 6.62240346253192063301188635000, 9.033454260657084619798557875227, 9.961610717092506640544663989415, 11.02903033371589609284762008626, 11.91672281736447066750399824000, 12.90920310871747642349251331801

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