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Properties

Label 2-120-15.14-c2-0-8
Degree $2$
Conductor $120$
Sign $-0.395 + 0.918i$
Analytic cond. $3.26976$
Root an. cond. $1.80824$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.49 + 1.67i)3-s + (−4.19 + 2.71i)5-s − 12.7i·7-s + (3.41 − 8.32i)9-s − 12.6i·11-s + 7.44i·13-s + (5.92 − 13.7i)15-s − 14.0·17-s − 31.0·19-s + (21.3 + 31.8i)21-s + 7.50·23-s + (10.2 − 22.7i)25-s + (5.40 + 26.4i)27-s − 15.7i·29-s − 20.4·31-s + ⋯
L(s)  = 1  + (−0.830 + 0.557i)3-s + (−0.839 + 0.542i)5-s − 1.82i·7-s + (0.379 − 0.925i)9-s − 1.14i·11-s + 0.572i·13-s + (0.395 − 0.918i)15-s − 0.826·17-s − 1.63·19-s + (1.01 + 1.51i)21-s + 0.326·23-s + (0.410 − 0.911i)25-s + (0.200 + 0.979i)27-s − 0.542i·29-s − 0.660·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.395 + 0.918i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.395 + 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(120\)    =    \(2^{3} \cdot 3 \cdot 5\)
Sign: $-0.395 + 0.918i$
Analytic conductor: \(3.26976\)
Root analytic conductor: \(1.80824\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{120} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 120,\ (\ :1),\ -0.395 + 0.918i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.250669 - 0.380661i\)
\(L(\frac12)\) \(\approx\) \(0.250669 - 0.380661i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (2.49 - 1.67i)T \)
5 \( 1 + (4.19 - 2.71i)T \)
good7 \( 1 + 12.7iT - 49T^{2} \)
11 \( 1 + 12.6iT - 121T^{2} \)
13 \( 1 - 7.44iT - 169T^{2} \)
17 \( 1 + 14.0T + 289T^{2} \)
19 \( 1 + 31.0T + 361T^{2} \)
23 \( 1 - 7.50T + 529T^{2} \)
29 \( 1 + 15.7iT - 841T^{2} \)
31 \( 1 + 20.4T + 961T^{2} \)
37 \( 1 - 12.9iT - 1.36e3T^{2} \)
41 \( 1 - 13.8iT - 1.68e3T^{2} \)
43 \( 1 + 30.0iT - 1.84e3T^{2} \)
47 \( 1 - 20.2T + 2.20e3T^{2} \)
53 \( 1 - 29.1T + 2.80e3T^{2} \)
59 \( 1 + 47.6iT - 3.48e3T^{2} \)
61 \( 1 - 43.0T + 3.72e3T^{2} \)
67 \( 1 + 0.630iT - 4.48e3T^{2} \)
71 \( 1 - 90.4iT - 5.04e3T^{2} \)
73 \( 1 + 46.2iT - 5.32e3T^{2} \)
79 \( 1 - 37.9T + 6.24e3T^{2} \)
83 \( 1 + 80.2T + 6.88e3T^{2} \)
89 \( 1 + 140. iT - 7.92e3T^{2} \)
97 \( 1 - 10.3iT - 9.40e3T^{2} \)
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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

−12.90293608568334420173002610959, −11.40168670965986017188139244271, −10.96133037848625659301737886181, −10.19712033657562920363710530620, −8.592447607602660700532630755298, −7.16291649795044433893661150168, −6.36814284473545897514082468923, −4.44160569299470666691247827344, −3.72234356438471816557325165737, −0.34581136731418819330225097158, 2.16994042765668226955679035734, 4.58382848504737438580459639600, 5.62129670162224235693627157669, 6.90213191318128352761116042000, 8.201424434673593367859298464926, 9.118521527512379496082023127640, 10.74767734096774475222179178666, 11.74396774112277021947539156877, 12.59713872140393351725350618621, 12.86163479146174888118084722039

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