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Properties

Label 2-120-24.11-c3-0-29
Degree $2$
Conductor $120$
Sign $0.832 + 0.553i$
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  + (−2.03 + 1.96i)2-s + (5.02 − 1.30i)3-s + (0.292 − 7.99i)4-s − 5·5-s + (−7.68 + 12.5i)6-s − 8.16i·7-s + (15.0 + 16.8i)8-s + (23.6 − 13.1i)9-s + (10.1 − 9.81i)10-s − 16.0i·11-s + (−8.95 − 40.5i)12-s − 41.4i·13-s + (16.0 + 16.6i)14-s + (−25.1 + 6.51i)15-s + (−63.8 − 4.67i)16-s − 83.3i·17-s + ⋯
L(s)  = 1  + (−0.719 + 0.694i)2-s + (0.968 − 0.250i)3-s + (0.0365 − 0.999i)4-s − 0.447·5-s + (−0.522 + 0.852i)6-s − 0.440i·7-s + (0.667 + 0.744i)8-s + (0.874 − 0.485i)9-s + (0.321 − 0.310i)10-s − 0.438i·11-s + (−0.215 − 0.976i)12-s − 0.884i·13-s + (0.305 + 0.317i)14-s + (−0.432 + 0.112i)15-s + (−0.997 − 0.0730i)16-s − 1.18i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.553i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.832 + 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(120\)    =    \(2^{3} \cdot 3 \cdot 5\)
Sign: $0.832 + 0.553i$
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: \((2,\ 120,\ (\ :3/2),\ 0.832 + 0.553i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.33438 - 0.402990i\)
\(L(\frac12)\) \(\approx\) \(1.33438 - 0.402990i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.03 - 1.96i)T \)
3 \( 1 + (-5.02 + 1.30i)T \)
5 \( 1 + 5T \)
good7 \( 1 + 8.16iT - 343T^{2} \)
11 \( 1 + 16.0iT - 1.33e3T^{2} \)
13 \( 1 + 41.4iT - 2.19e3T^{2} \)
17 \( 1 + 83.3iT - 4.91e3T^{2} \)
19 \( 1 - 54.9T + 6.85e3T^{2} \)
23 \( 1 - 66.4T + 1.21e4T^{2} \)
29 \( 1 - 153.T + 2.43e4T^{2} \)
31 \( 1 + 11.4iT - 2.97e4T^{2} \)
37 \( 1 - 245. iT - 5.06e4T^{2} \)
41 \( 1 - 14.3iT - 6.89e4T^{2} \)
43 \( 1 + 485.T + 7.95e4T^{2} \)
47 \( 1 + 70.6T + 1.03e5T^{2} \)
53 \( 1 + 514.T + 1.48e5T^{2} \)
59 \( 1 + 488. iT - 2.05e5T^{2} \)
61 \( 1 - 886. iT - 2.26e5T^{2} \)
67 \( 1 + 780.T + 3.00e5T^{2} \)
71 \( 1 - 520.T + 3.57e5T^{2} \)
73 \( 1 - 490.T + 3.89e5T^{2} \)
79 \( 1 + 1.15e3iT - 4.93e5T^{2} \)
83 \( 1 - 1.10e3iT - 5.71e5T^{2} \)
89 \( 1 - 1.24e3iT - 7.04e5T^{2} \)
97 \( 1 - 296.T + 9.12e5T^{2} \)
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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

−13.32375436153599823683244987417, −11.77909620161942781546604840319, −10.46574512994604544462091213422, −9.493915428346068138947027039380, −8.407451706107925473246175675195, −7.63005152036219100548461514351, −6.67860249462768773897857195248, −4.92671028997821210753326258743, −3.06638692773878948863279542587, −0.909429074821368537236335489681, 1.80468504670211722472699668152, 3.26633622879587546495861172092, 4.48662905739291295787239652333, 6.93342313782748672876817498697, 8.064153045785308325842732671140, 8.899202398758351584037932985736, 9.794586576170469682862373139023, 10.86303477099375122478051862371, 12.03659117162090400613546449427, 12.88660897521641786925755899922

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