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Properties

Label 2-120-24.11-c3-0-10
Degree $2$
Conductor $120$
Sign $0.981 - 0.192i$
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  + (−2.40 − 1.49i)2-s + (−1.49 − 4.97i)3-s + (3.53 + 7.17i)4-s + 5·5-s + (−3.84 + 14.1i)6-s + 14.5i·7-s + (2.21 − 22.5i)8-s + (−22.5 + 14.8i)9-s + (−12.0 − 7.46i)10-s + 58.0i·11-s + (30.4 − 28.3i)12-s + 43.9i·13-s + (21.7 − 35.0i)14-s + (−7.46 − 24.8i)15-s + (−38.9 + 50.7i)16-s − 78.8i·17-s + ⋯
L(s)  = 1  + (−0.849 − 0.528i)2-s + (−0.287 − 0.957i)3-s + (0.442 + 0.896i)4-s + 0.447·5-s + (−0.261 + 0.965i)6-s + 0.788i·7-s + (0.0978 − 0.995i)8-s + (−0.834 + 0.550i)9-s + (−0.379 − 0.236i)10-s + 1.59i·11-s + (0.731 − 0.681i)12-s + 0.937i·13-s + (0.416 − 0.669i)14-s + (−0.128 − 0.428i)15-s + (−0.608 + 0.793i)16-s − 1.12i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.192i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.981 - 0.192i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(120\)    =    \(2^{3} \cdot 3 \cdot 5\)
Sign: $0.981 - 0.192i$
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: \((2,\ 120,\ (\ :3/2),\ 0.981 - 0.192i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.913440 + 0.0886495i\)
\(L(\frac12)\) \(\approx\) \(0.913440 + 0.0886495i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.40 + 1.49i)T \)
3 \( 1 + (1.49 + 4.97i)T \)
5 \( 1 - 5T \)
good7 \( 1 - 14.5iT - 343T^{2} \)
11 \( 1 - 58.0iT - 1.33e3T^{2} \)
13 \( 1 - 43.9iT - 2.19e3T^{2} \)
17 \( 1 + 78.8iT - 4.91e3T^{2} \)
19 \( 1 - 103.T + 6.85e3T^{2} \)
23 \( 1 - 163.T + 1.21e4T^{2} \)
29 \( 1 + 247.T + 2.43e4T^{2} \)
31 \( 1 - 153. iT - 2.97e4T^{2} \)
37 \( 1 - 153. iT - 5.06e4T^{2} \)
41 \( 1 + 27.0iT - 6.89e4T^{2} \)
43 \( 1 - 246.T + 7.95e4T^{2} \)
47 \( 1 + 411.T + 1.03e5T^{2} \)
53 \( 1 + 80.1T + 1.48e5T^{2} \)
59 \( 1 - 513. iT - 2.05e5T^{2} \)
61 \( 1 - 27.0iT - 2.26e5T^{2} \)
67 \( 1 + 453.T + 3.00e5T^{2} \)
71 \( 1 - 446.T + 3.57e5T^{2} \)
73 \( 1 - 524.T + 3.89e5T^{2} \)
79 \( 1 + 93.4iT - 4.93e5T^{2} \)
83 \( 1 - 1.24e3iT - 5.71e5T^{2} \)
89 \( 1 - 267. iT - 7.04e5T^{2} \)
97 \( 1 + 897.T + 9.12e5T^{2} \)
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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

−12.68027213059013263733861471397, −11.96182281902225148749661408095, −11.16735909335963690765622953898, −9.635209362447710428237430681791, −9.012868264496360809275917223302, −7.45684728832248527794618139028, −6.83024933939881252825533159184, −5.12953556800879866073644373965, −2.67599215308478971877669377417, −1.51080657944727960287609019493, 0.71046628285823037498863276096, 3.38686288064338232205202491401, 5.32606688320603753079104783300, 6.10559946811633329514708512131, 7.68120342385582159469941414535, 8.838078269607522073909015099174, 9.767904593336959764661395019628, 10.80965330112808755079987314653, 11.20593447444800469681710329954, 13.20088415709506303232332557215

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