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Properties

Label 2-120-24.11-c3-0-31
Degree $2$
Conductor $120$
Sign $-0.765 + 0.642i$
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  + (−2.65 + 0.968i)2-s + (−4.88 − 1.78i)3-s + (6.12 − 5.14i)4-s + 5·5-s + (14.6 + 0.0110i)6-s − 12.1i·7-s + (−11.2 + 19.6i)8-s + (20.6 + 17.4i)9-s + (−13.2 + 4.84i)10-s + 47.5i·11-s + (−39.0 + 14.2i)12-s − 55.3i·13-s + (11.8 + 32.3i)14-s + (−24.4 − 8.91i)15-s + (11.0 − 63.0i)16-s − 90.1i·17-s + ⋯
L(s)  = 1  + (−0.939 + 0.342i)2-s + (−0.939 − 0.343i)3-s + (0.765 − 0.643i)4-s + 0.447·5-s + (0.999 + 0.000752i)6-s − 0.658i·7-s + (−0.498 + 0.866i)8-s + (0.764 + 0.644i)9-s + (−0.420 + 0.153i)10-s + 1.30i·11-s + (−0.939 + 0.341i)12-s − 1.17i·13-s + (0.225 + 0.618i)14-s + (−0.420 − 0.153i)15-s + (0.171 − 0.985i)16-s − 1.28i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.765 + 0.642i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.765 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(120\)    =    \(2^{3} \cdot 3 \cdot 5\)
Sign: $-0.765 + 0.642i$
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: \((2,\ 120,\ (\ :3/2),\ -0.765 + 0.642i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.111361 - 0.305917i\)
\(L(\frac12)\) \(\approx\) \(0.111361 - 0.305917i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.65 - 0.968i)T \)
3 \( 1 + (4.88 + 1.78i)T \)
5 \( 1 - 5T \)
good7 \( 1 + 12.1iT - 343T^{2} \)
11 \( 1 - 47.5iT - 1.33e3T^{2} \)
13 \( 1 + 55.3iT - 2.19e3T^{2} \)
17 \( 1 + 90.1iT - 4.91e3T^{2} \)
19 \( 1 + 123.T + 6.85e3T^{2} \)
23 \( 1 + 136.T + 1.21e4T^{2} \)
29 \( 1 + 122.T + 2.43e4T^{2} \)
31 \( 1 + 217. iT - 2.97e4T^{2} \)
37 \( 1 - 359. iT - 5.06e4T^{2} \)
41 \( 1 + 158. iT - 6.89e4T^{2} \)
43 \( 1 + 334.T + 7.95e4T^{2} \)
47 \( 1 + 55.8T + 1.03e5T^{2} \)
53 \( 1 + 494.T + 1.48e5T^{2} \)
59 \( 1 + 151. iT - 2.05e5T^{2} \)
61 \( 1 + 316. iT - 2.26e5T^{2} \)
67 \( 1 - 460.T + 3.00e5T^{2} \)
71 \( 1 - 55.3T + 3.57e5T^{2} \)
73 \( 1 + 284.T + 3.89e5T^{2} \)
79 \( 1 + 374. iT - 4.93e5T^{2} \)
83 \( 1 + 561. iT - 5.71e5T^{2} \)
89 \( 1 + 3.39iT - 7.04e5T^{2} \)
97 \( 1 - 1.04e3T + 9.12e5T^{2} \)
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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

−12.45328811579319238367635074130, −11.33855163579993078095122957037, −10.27259367615834812058644858782, −9.766742541804391139970894312586, −8.003216131097939937932050173420, −7.11184877448681091439266652914, −6.10637571764979324746910497359, −4.82607752831013463184032514958, −1.97966612919560627622855022458, −0.24950432964218182053765422770, 1.82211919373258666912203205833, 3.89118355746188686947977023216, 5.87373145950356492692298721912, 6.57419165595827384804054192626, 8.389019069361261599138251979784, 9.222804624670522662663179830983, 10.41979205396524684545512350449, 11.11423628358669506283739952151, 12.07457036899214205744265056057, 12.96934225729291430217998532271

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