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Properties

Label 2-120-15.14-c2-0-9
Degree $2$
Conductor $120$
Sign $0.105 + 0.994i$
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  + (−0.938 − 2.84i)3-s + (4.88 + 1.05i)5-s − 6.81i·7-s + (−7.23 + 5.34i)9-s − 7.52i·11-s − 16.2i·13-s + (−1.58 − 14.9i)15-s + 4.11·17-s + 7.86·19-s + (−19.4 + 6.39i)21-s − 19.5·23-s + (22.7 + 10.2i)25-s + (22.0 + 15.6i)27-s + 55.8i·29-s + 43.4·31-s + ⋯
L(s)  = 1  + (−0.312 − 0.949i)3-s + (0.977 + 0.210i)5-s − 0.973i·7-s + (−0.804 + 0.594i)9-s − 0.684i·11-s − 1.24i·13-s + (−0.105 − 0.994i)15-s + 0.242·17-s + 0.413·19-s + (−0.924 + 0.304i)21-s − 0.848·23-s + (0.911 + 0.411i)25-s + (0.815 + 0.578i)27-s + 1.92i·29-s + 1.40·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.105 + 0.994i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.105 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(120\)    =    \(2^{3} \cdot 3 \cdot 5\)
Sign: $0.105 + 0.994i$
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: \((2,\ 120,\ (\ :1),\ 0.105 + 0.994i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.01976 - 0.916953i\)
\(L(\frac12)\) \(\approx\) \(1.01976 - 0.916953i\)
\(L(2)\) 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 \( 1 + (0.938 + 2.84i)T \)
5 \( 1 + (-4.88 - 1.05i)T \)
good7 \( 1 + 6.81iT - 49T^{2} \)
11 \( 1 + 7.52iT - 121T^{2} \)
13 \( 1 + 16.2iT - 169T^{2} \)
17 \( 1 - 4.11T + 289T^{2} \)
19 \( 1 - 7.86T + 361T^{2} \)
23 \( 1 + 19.5T + 529T^{2} \)
29 \( 1 - 55.8iT - 841T^{2} \)
31 \( 1 - 43.4T + 961T^{2} \)
37 \( 1 - 31.5iT - 1.36e3T^{2} \)
41 \( 1 + 51.3iT - 1.68e3T^{2} \)
43 \( 1 - 51.2iT - 1.84e3T^{2} \)
47 \( 1 + 61.7T + 2.20e3T^{2} \)
53 \( 1 - 82.7T + 2.80e3T^{2} \)
59 \( 1 - 97.6iT - 3.48e3T^{2} \)
61 \( 1 - 4.13T + 3.72e3T^{2} \)
67 \( 1 + 63.1iT - 4.48e3T^{2} \)
71 \( 1 - 40.3iT - 5.04e3T^{2} \)
73 \( 1 + 78.5iT - 5.32e3T^{2} \)
79 \( 1 + 51.0T + 6.24e3T^{2} \)
83 \( 1 - 2.72T + 6.88e3T^{2} \)
89 \( 1 + 70.4iT - 7.92e3T^{2} \)
97 \( 1 + 3.44iT - 9.40e3T^{2} \)
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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

−13.23324441273156994379058004215, −12.10473944682853263616577233247, −10.79725310236103917817010500943, −10.13741146730341751807772425902, −8.507012171003124166637100259976, −7.39926417855777764469486198968, −6.30688307067651636969606362297, −5.28741834642590381773301719192, −3.02563573987946303984327545999, −1.11392599333177603445135564541, 2.32326206936499089736232342974, 4.33413842706774261072523630098, 5.51239790764811655943280810657, 6.46779109570216492178391704951, 8.461004314567040366799733637155, 9.564803864254695080905793075935, 9.969470847405976305904849308277, 11.52518335117122779580004310382, 12.19527423327649653005578888243, 13.62647396333217442376355210456

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