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Properties

Label 2-120-120.53-c3-0-39
Degree $2$
Conductor $120$
Sign $0.988 + 0.154i$
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  + (−0.401 + 2.79i)2-s + (−2.36 + 4.62i)3-s + (−7.67 − 2.24i)4-s + (3.00 − 10.7i)5-s + (−12.0 − 8.48i)6-s + (13.4 − 13.4i)7-s + (9.37 − 20.5i)8-s + (−15.8 − 21.8i)9-s + (28.9 + 12.7i)10-s − 30.2·11-s + (28.5 − 30.2i)12-s + (25.9 − 25.9i)13-s + (32.2 + 43.0i)14-s + (42.7 + 39.3i)15-s + (53.8 + 34.5i)16-s + (−26.6 − 26.6i)17-s + ⋯
L(s)  = 1  + (−0.141 + 0.989i)2-s + (−0.455 + 0.890i)3-s + (−0.959 − 0.281i)4-s + (0.268 − 0.963i)5-s + (−0.816 − 0.577i)6-s + (0.725 − 0.725i)7-s + (0.414 − 0.910i)8-s + (−0.585 − 0.810i)9-s + (0.915 + 0.402i)10-s − 0.829·11-s + (0.687 − 0.726i)12-s + (0.553 − 0.553i)13-s + (0.614 + 0.820i)14-s + (0.735 + 0.677i)15-s + (0.841 + 0.539i)16-s + (−0.380 − 0.380i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.154i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.988 + 0.154i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(120\)    =    \(2^{3} \cdot 3 \cdot 5\)
Sign: $0.988 + 0.154i$
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: \((2,\ 120,\ (\ :3/2),\ 0.988 + 0.154i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.00132 - 0.0776762i\)
\(L(\frac12)\) \(\approx\) \(1.00132 - 0.0776762i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.401 - 2.79i)T \)
3 \( 1 + (2.36 - 4.62i)T \)
5 \( 1 + (-3.00 + 10.7i)T \)
good7 \( 1 + (-13.4 + 13.4i)T - 343iT^{2} \)
11 \( 1 + 30.2T + 1.33e3T^{2} \)
13 \( 1 + (-25.9 + 25.9i)T - 2.19e3iT^{2} \)
17 \( 1 + (26.6 + 26.6i)T + 4.91e3iT^{2} \)
19 \( 1 - 48.6T + 6.85e3T^{2} \)
23 \( 1 + (59.4 - 59.4i)T - 1.21e4iT^{2} \)
29 \( 1 + 254. iT - 2.43e4T^{2} \)
31 \( 1 - 240.T + 2.97e4T^{2} \)
37 \( 1 + (177. + 177. i)T + 5.06e4iT^{2} \)
41 \( 1 - 88.8iT - 6.89e4T^{2} \)
43 \( 1 + (-335. + 335. i)T - 7.95e4iT^{2} \)
47 \( 1 + (316. + 316. i)T + 1.03e5iT^{2} \)
53 \( 1 + (-53.4 - 53.4i)T + 1.48e5iT^{2} \)
59 \( 1 - 26.0iT - 2.05e5T^{2} \)
61 \( 1 - 685. iT - 2.26e5T^{2} \)
67 \( 1 + (429. + 429. i)T + 3.00e5iT^{2} \)
71 \( 1 - 2.66iT - 3.57e5T^{2} \)
73 \( 1 + (417. + 417. i)T + 3.89e5iT^{2} \)
79 \( 1 - 662. iT - 4.93e5T^{2} \)
83 \( 1 + (-915. - 915. i)T + 5.71e5iT^{2} \)
89 \( 1 + 470.T + 7.04e5T^{2} \)
97 \( 1 + (-1.20e3 + 1.20e3i)T - 9.12e5iT^{2} \)
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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

−13.37380165393008807997187366122, −11.92044008621227347848880714622, −10.56659475666374944000192864059, −9.742626145513208592919144320902, −8.600208441192221651434222512809, −7.67556532289017712176786447784, −5.93610327235817686342233306186, −5.06189965554009759426638603041, −4.10843056656529638414291803472, −0.62116183402971448224425973737, 1.69318367658863934971517454210, 2.86297884231410036436295397140, 4.97807956183264880571556483675, 6.26683185797364845688477754988, 7.74613020052404992431146502098, 8.721458172591559297055762308116, 10.28341194029325590778427553065, 11.11510483144459059863582917891, 11.81735957079665405820898037727, 12.84818726797171415670495408168

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