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Properties

Label 2-120-120.107-c2-0-36
Degree $2$
Conductor $120$
Sign $0.178 + 0.983i$
Analytic cond. $3.26976$
Root an. cond. $1.80824$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.952 − 1.75i)2-s + (2.40 + 1.79i)3-s + (−2.18 − 3.34i)4-s + (−0.927 − 4.91i)5-s + (5.45 − 2.50i)6-s + (3.37 − 3.37i)7-s + (−7.97 + 0.658i)8-s + (2.52 + 8.63i)9-s + (−9.52 − 3.04i)10-s − 11.5i·11-s + (0.776 − 11.9i)12-s + (14.6 + 14.6i)13-s + (−2.72 − 9.15i)14-s + (6.61 − 13.4i)15-s + (−6.43 + 14.6i)16-s + (3.42 − 3.42i)17-s + ⋯
L(s)  = 1  + (0.476 − 0.879i)2-s + (0.800 + 0.599i)3-s + (−0.546 − 0.837i)4-s + (−0.185 − 0.982i)5-s + (0.908 − 0.418i)6-s + (0.482 − 0.482i)7-s + (−0.996 + 0.0823i)8-s + (0.280 + 0.959i)9-s + (−0.952 − 0.304i)10-s − 1.04i·11-s + (0.0647 − 0.997i)12-s + (1.12 + 1.12i)13-s + (−0.194 − 0.654i)14-s + (0.441 − 0.897i)15-s + (−0.402 + 0.915i)16-s + (0.201 − 0.201i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.178 + 0.983i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.178 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(120\)    =    \(2^{3} \cdot 3 \cdot 5\)
Sign: $0.178 + 0.983i$
Analytic conductor: \(3.26976\)
Root analytic conductor: \(1.80824\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{120} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 120,\ (\ :1),\ 0.178 + 0.983i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.59638 - 1.33312i\)
\(L(\frac12)\) \(\approx\) \(1.59638 - 1.33312i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.952 + 1.75i)T \)
3 \( 1 + (-2.40 - 1.79i)T \)
5 \( 1 + (0.927 + 4.91i)T \)
good7 \( 1 + (-3.37 + 3.37i)T - 49iT^{2} \)
11 \( 1 + 11.5iT - 121T^{2} \)
13 \( 1 + (-14.6 - 14.6i)T + 169iT^{2} \)
17 \( 1 + (-3.42 + 3.42i)T - 289iT^{2} \)
19 \( 1 - 10.0iT - 361T^{2} \)
23 \( 1 + (23.7 - 23.7i)T - 529iT^{2} \)
29 \( 1 + 16.1iT - 841T^{2} \)
31 \( 1 - 15.4iT - 961T^{2} \)
37 \( 1 + (-21.8 + 21.8i)T - 1.36e3iT^{2} \)
41 \( 1 - 52.7iT - 1.68e3T^{2} \)
43 \( 1 + (1.63 - 1.63i)T - 1.84e3iT^{2} \)
47 \( 1 + (17.0 + 17.0i)T + 2.20e3iT^{2} \)
53 \( 1 + (-57.1 + 57.1i)T - 2.80e3iT^{2} \)
59 \( 1 - 65.3T + 3.48e3T^{2} \)
61 \( 1 - 19.8iT - 3.72e3T^{2} \)
67 \( 1 + (89.6 + 89.6i)T + 4.48e3iT^{2} \)
71 \( 1 + 89.6T + 5.04e3T^{2} \)
73 \( 1 + (-48.6 + 48.6i)T - 5.32e3iT^{2} \)
79 \( 1 + 74.9T + 6.24e3T^{2} \)
83 \( 1 + (21.0 + 21.0i)T + 6.88e3iT^{2} \)
89 \( 1 - 46.5T + 7.92e3T^{2} \)
97 \( 1 + (-21.3 - 21.3i)T + 9.40e3iT^{2} \)
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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

−13.37637652759126278504667707185, −11.82115276860139597388235001501, −11.08149752993525738531248763796, −9.849956941185452798375868280996, −8.898871303675966270607795047457, −8.069295447005937260931493698666, −5.77144411389446440262518141468, −4.40970205242712987432437019323, −3.60599918407414710945310034316, −1.53903126290133643215786225682, 2.62906584795493710106278114792, 4.00916091423453938920262667589, 5.85971334219807460177366490773, 6.95513908168956672271453886592, 7.893227412104136036579930521521, 8.712274051077300680910388370548, 10.20540076281759709472134361112, 11.77323287046265876520456209691, 12.73117434571715526511228946294, 13.64571055607868088848646802817

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