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Properties

Label 2-120-24.11-c3-0-42
Degree $2$
Conductor $120$
Sign $0.970 + 0.241i$
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  + (2.78 + 0.487i)2-s + (4.99 − 1.41i)3-s + (7.52 + 2.71i)4-s − 5·5-s + (14.6 − 1.50i)6-s − 35.4i·7-s + (19.6 + 11.2i)8-s + (22.9 − 14.1i)9-s + (−13.9 − 2.43i)10-s + 20.0i·11-s + (41.4 + 2.93i)12-s + 45.5i·13-s + (17.2 − 98.7i)14-s + (−24.9 + 7.07i)15-s + (49.2 + 40.8i)16-s + 103. i·17-s + ⋯
L(s)  = 1  + (0.985 + 0.172i)2-s + (0.962 − 0.272i)3-s + (0.940 + 0.339i)4-s − 0.447·5-s + (0.994 − 0.102i)6-s − 1.91i·7-s + (0.868 + 0.496i)8-s + (0.851 − 0.523i)9-s + (−0.440 − 0.0770i)10-s + 0.550i·11-s + (0.997 + 0.0705i)12-s + 0.971i·13-s + (0.329 − 1.88i)14-s + (−0.430 + 0.121i)15-s + (0.769 + 0.638i)16-s + 1.47i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.241i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.970 + 0.241i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(120\)    =    \(2^{3} \cdot 3 \cdot 5\)
Sign: $0.970 + 0.241i$
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: \((2,\ 120,\ (\ :3/2),\ 0.970 + 0.241i)\)

Particular Values

\(L(2)\) \(\approx\) \(3.60491 - 0.441460i\)
\(L(\frac12)\) \(\approx\) \(3.60491 - 0.441460i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.78 - 0.487i)T \)
3 \( 1 + (-4.99 + 1.41i)T \)
5 \( 1 + 5T \)
good7 \( 1 + 35.4iT - 343T^{2} \)
11 \( 1 - 20.0iT - 1.33e3T^{2} \)
13 \( 1 - 45.5iT - 2.19e3T^{2} \)
17 \( 1 - 103. iT - 4.91e3T^{2} \)
19 \( 1 + 43.2T + 6.85e3T^{2} \)
23 \( 1 + 149.T + 1.21e4T^{2} \)
29 \( 1 + 72.3T + 2.43e4T^{2} \)
31 \( 1 + 59.8iT - 2.97e4T^{2} \)
37 \( 1 - 29.9iT - 5.06e4T^{2} \)
41 \( 1 + 289. iT - 6.89e4T^{2} \)
43 \( 1 - 157.T + 7.95e4T^{2} \)
47 \( 1 - 17.9T + 1.03e5T^{2} \)
53 \( 1 + 93.1T + 1.48e5T^{2} \)
59 \( 1 + 580. iT - 2.05e5T^{2} \)
61 \( 1 - 548. iT - 2.26e5T^{2} \)
67 \( 1 - 556.T + 3.00e5T^{2} \)
71 \( 1 - 566.T + 3.57e5T^{2} \)
73 \( 1 + 913.T + 3.89e5T^{2} \)
79 \( 1 - 307. iT - 4.93e5T^{2} \)
83 \( 1 - 1.09e3iT - 5.71e5T^{2} \)
89 \( 1 - 72.0iT - 7.04e5T^{2} \)
97 \( 1 + 888.T + 9.12e5T^{2} \)
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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

−13.16456782846880557260865963148, −12.33074315843313942166642503068, −10.95551772206222333452563590757, −9.995988582398643643317849452900, −8.205167323563096292457006410575, −7.36853199521894433865482542915, −6.53311042322441354712370708704, −4.16903422396331722013568009498, −3.89569577944416332877513540707, −1.80938744127688492960446571997, 2.36052394523087306357171960450, 3.28014960061669778146328339604, 4.88378303318359843101268165538, 6.00307279728942269702477541413, 7.67859918497678111231844604940, 8.704911651722923788307781195836, 9.874754791060415669607605250201, 11.27292015670384957209782508937, 12.19574507365089989914162671964, 13.02489639246856249089882650753

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