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Properties

Label 2-120-24.11-c3-0-7
Degree $2$
Conductor $120$
Sign $-0.873 + 0.487i$
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  + (0.638 + 2.75i)2-s + (−0.899 + 5.11i)3-s + (−7.18 + 3.51i)4-s + 5·5-s + (−14.6 + 0.787i)6-s + 23.1i·7-s + (−14.2 − 17.5i)8-s + (−25.3 − 9.21i)9-s + (3.19 + 13.7i)10-s + 5.98i·11-s + (−11.5 − 39.9i)12-s − 3.38i·13-s + (−63.7 + 14.7i)14-s + (−4.49 + 25.5i)15-s + (39.2 − 50.5i)16-s − 35.8i·17-s + ⋯
L(s)  = 1  + (0.225 + 0.974i)2-s + (−0.173 + 0.984i)3-s + (−0.898 + 0.439i)4-s + 0.447·5-s + (−0.998 + 0.0535i)6-s + 1.24i·7-s + (−0.631 − 0.775i)8-s + (−0.940 − 0.341i)9-s + (0.100 + 0.435i)10-s + 0.163i·11-s + (−0.277 − 0.960i)12-s − 0.0722i·13-s + (−1.21 + 0.281i)14-s + (−0.0774 + 0.440i)15-s + (0.613 − 0.789i)16-s − 0.512i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.873 + 0.487i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.873 + 0.487i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(120\)    =    \(2^{3} \cdot 3 \cdot 5\)
Sign: $-0.873 + 0.487i$
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: \((2,\ 120,\ (\ :3/2),\ -0.873 + 0.487i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.316551 - 1.21695i\)
\(L(\frac12)\) \(\approx\) \(0.316551 - 1.21695i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.638 - 2.75i)T \)
3 \( 1 + (0.899 - 5.11i)T \)
5 \( 1 - 5T \)
good7 \( 1 - 23.1iT - 343T^{2} \)
11 \( 1 - 5.98iT - 1.33e3T^{2} \)
13 \( 1 + 3.38iT - 2.19e3T^{2} \)
17 \( 1 + 35.8iT - 4.91e3T^{2} \)
19 \( 1 + 51.7T + 6.85e3T^{2} \)
23 \( 1 - 123.T + 1.21e4T^{2} \)
29 \( 1 + 130.T + 2.43e4T^{2} \)
31 \( 1 - 298. iT - 2.97e4T^{2} \)
37 \( 1 - 343. iT - 5.06e4T^{2} \)
41 \( 1 - 121. iT - 6.89e4T^{2} \)
43 \( 1 + 535.T + 7.95e4T^{2} \)
47 \( 1 - 401.T + 1.03e5T^{2} \)
53 \( 1 - 67.6T + 1.48e5T^{2} \)
59 \( 1 - 717. iT - 2.05e5T^{2} \)
61 \( 1 - 45.2iT - 2.26e5T^{2} \)
67 \( 1 - 722.T + 3.00e5T^{2} \)
71 \( 1 - 1.03e3T + 3.57e5T^{2} \)
73 \( 1 + 331.T + 3.89e5T^{2} \)
79 \( 1 + 449. iT - 4.93e5T^{2} \)
83 \( 1 + 641. iT - 5.71e5T^{2} \)
89 \( 1 + 855. iT - 7.04e5T^{2} \)
97 \( 1 - 82.0T + 9.12e5T^{2} \)
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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

−13.86634996931827260098239754857, −12.68532409230088315644882754728, −11.64086221779685681012342225501, −10.17870873224572202348829667568, −9.129836574622906474432691260652, −8.515649288189798112038251013200, −6.72734216137247313282131393563, −5.57155964515832359180738163365, −4.80552946114878446297133495068, −3.07540505003977093555547839956, 0.63889188177675695468323472960, 2.09428145532602913218891926176, 3.83605703817100966456683691639, 5.44938637935029965264677068785, 6.76291275634226447388649859774, 8.084010662566168874587158012537, 9.389571780102453789475666481958, 10.67345009693518578732712729846, 11.26660529430673709400403283519, 12.62433424868331429199768616847

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