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Properties

Label 2-120-24.11-c3-0-22
Degree $2$
Conductor $120$
Sign $0.524 - 0.851i$
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  + (2.81 − 0.288i)2-s + (−3.93 + 3.39i)3-s + (7.83 − 1.62i)4-s + 5·5-s + (−10.0 + 10.6i)6-s + 20.9i·7-s + (21.5 − 6.83i)8-s + (3.93 − 26.7i)9-s + (14.0 − 1.44i)10-s + 46.5i·11-s + (−25.2 + 32.9i)12-s + 14.8i·13-s + (6.03 + 58.8i)14-s + (−19.6 + 16.9i)15-s + (58.7 − 25.4i)16-s + 19.4i·17-s + ⋯
L(s)  = 1  + (0.994 − 0.102i)2-s + (−0.756 + 0.653i)3-s + (0.979 − 0.203i)4-s + 0.447·5-s + (−0.686 + 0.727i)6-s + 1.12i·7-s + (0.953 − 0.301i)8-s + (0.145 − 0.989i)9-s + (0.444 − 0.0456i)10-s + 1.27i·11-s + (−0.608 + 0.793i)12-s + 0.315i·13-s + (0.115 + 1.12i)14-s + (−0.338 + 0.292i)15-s + (0.917 − 0.397i)16-s + 0.277i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.524 - 0.851i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.524 - 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(120\)    =    \(2^{3} \cdot 3 \cdot 5\)
Sign: $0.524 - 0.851i$
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: \((2,\ 120,\ (\ :3/2),\ 0.524 - 0.851i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.19347 + 1.22553i\)
\(L(\frac12)\) \(\approx\) \(2.19347 + 1.22553i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.81 + 0.288i)T \)
3 \( 1 + (3.93 - 3.39i)T \)
5 \( 1 - 5T \)
good7 \( 1 - 20.9iT - 343T^{2} \)
11 \( 1 - 46.5iT - 1.33e3T^{2} \)
13 \( 1 - 14.8iT - 2.19e3T^{2} \)
17 \( 1 - 19.4iT - 4.91e3T^{2} \)
19 \( 1 - 120.T + 6.85e3T^{2} \)
23 \( 1 + 89.5T + 1.21e4T^{2} \)
29 \( 1 + 150.T + 2.43e4T^{2} \)
31 \( 1 + 332. iT - 2.97e4T^{2} \)
37 \( 1 + 119. iT - 5.06e4T^{2} \)
41 \( 1 + 516. iT - 6.89e4T^{2} \)
43 \( 1 - 35.5T + 7.95e4T^{2} \)
47 \( 1 + 288.T + 1.03e5T^{2} \)
53 \( 1 - 249.T + 1.48e5T^{2} \)
59 \( 1 - 9.27iT - 2.05e5T^{2} \)
61 \( 1 - 184. iT - 2.26e5T^{2} \)
67 \( 1 - 822.T + 3.00e5T^{2} \)
71 \( 1 + 578.T + 3.57e5T^{2} \)
73 \( 1 + 381.T + 3.89e5T^{2} \)
79 \( 1 - 991. iT - 4.93e5T^{2} \)
83 \( 1 + 227. iT - 5.71e5T^{2} \)
89 \( 1 + 401. iT - 7.04e5T^{2} \)
97 \( 1 - 1.45e3T + 9.12e5T^{2} \)
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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

−12.95038772488937709639718450543, −12.07899959033017793840070505808, −11.45680079162665083588925168996, −10.11637300645375839518068405070, −9.346531383003203898791074017019, −7.30062875442336548284245946577, −5.94834259230417542854348480549, −5.27597697076698885180309442805, −3.97108093026772291121951142167, −2.13922622622229937224526090239, 1.20445877369402169671128790541, 3.27683684642965989501232143415, 4.97867791843575410472327157249, 5.98532419715504931981492118476, 7.01663508236824225510609478572, 8.009095359620892401605389476686, 10.14499881224037617533521655371, 11.06998878541614397535215205314, 11.85400406280198940038660382993, 13.10188566878809242397567589722

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