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Properties

Label 2-120-24.11-c3-0-32
Degree $2$
Conductor $120$
Sign $0.700 + 0.713i$
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 + (−13.2 − 6.37i)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 + (33.7 + 24.2i)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 + (−0.900 − 0.433i)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 + (0.812 + 0.582i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.700 + 0.713i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.700 + 0.713i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(120\)    =    \(2^{3} \cdot 3 \cdot 5\)
Sign: $0.700 + 0.713i$
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: \((2,\ 120,\ (\ :3/2),\ 0.700 + 0.713i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.42795 - 0.599576i\)
\(L(\frac12)\) \(\approx\) \(1.42795 - 0.599576i\)
\(L(\frac{5}{2})\) 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

−13.16764270684813083947961115880, −11.39216974148710352912730588323, −10.50142253710123059015071220326, −9.657231524883634764453015509501, −8.771543545232624208971883683513, −7.47756568800224081159526894705, −6.81277815059319432631649849509, −4.35352066347888199617541121821, −2.91686972430134136289802206396, −1.11722166180437458209662371087, 1.82805534218347913207759367235, 2.87244056810020653899243650114, 5.53259946762221492199256899517, 6.66960352921925162028039685748, 8.144715760204826697407555208926, 8.766450758986009626461095436896, 9.615107396184240938916025095280, 10.73474372672419934251966511561, 12.33312204681860639898887948142, 12.81554935144577195346869776355

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