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Properties

Label 2-120-120.77-c1-0-0
Degree $2$
Conductor $120$
Sign $0.875 - 0.483i$
Analytic cond. $0.958204$
Root an. cond. $0.978879$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.30 − 0.533i)2-s + (−1.59 − 0.667i)3-s + (1.43 + 1.39i)4-s + (−0.143 + 2.23i)5-s + (1.73 + 1.72i)6-s + (0.582 + 0.582i)7-s + (−1.13 − 2.59i)8-s + (2.10 + 2.13i)9-s + (1.37 − 2.84i)10-s + 3.68·11-s + (−1.35 − 3.18i)12-s + (3.88 + 3.88i)13-s + (−0.452 − 1.07i)14-s + (1.71 − 3.47i)15-s + (0.0980 + 3.99i)16-s + (−0.880 + 0.880i)17-s + ⋯
L(s)  = 1  + (−0.926 − 0.377i)2-s + (−0.922 − 0.385i)3-s + (0.715 + 0.698i)4-s + (−0.0639 + 0.997i)5-s + (0.709 + 0.704i)6-s + (0.220 + 0.220i)7-s + (−0.399 − 0.916i)8-s + (0.703 + 0.711i)9-s + (0.435 − 0.900i)10-s + 1.11·11-s + (−0.391 − 0.920i)12-s + (1.07 + 1.07i)13-s + (−0.120 − 0.287i)14-s + (0.443 − 0.896i)15-s + (0.0245 + 0.999i)16-s + (−0.213 + 0.213i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.875 - 0.483i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.875 - 0.483i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(120\)    =    \(2^{3} \cdot 3 \cdot 5\)
Sign: $0.875 - 0.483i$
Analytic conductor: \(0.958204\)
Root analytic conductor: \(0.978879\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{120} (77, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 120,\ (\ :1/2),\ 0.875 - 0.483i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.547450 + 0.141291i\)
\(L(\frac12)\) \(\approx\) \(0.547450 + 0.141291i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.30 + 0.533i)T \)
3 \( 1 + (1.59 + 0.667i)T \)
5 \( 1 + (0.143 - 2.23i)T \)
good7 \( 1 + (-0.582 - 0.582i)T + 7iT^{2} \)
11 \( 1 - 3.68T + 11T^{2} \)
13 \( 1 + (-3.88 - 3.88i)T + 13iT^{2} \)
17 \( 1 + (0.880 - 0.880i)T - 17iT^{2} \)
19 \( 1 + 6.32T + 19T^{2} \)
23 \( 1 + (-2.06 - 2.06i)T + 23iT^{2} \)
29 \( 1 - 1.37iT - 29T^{2} \)
31 \( 1 - 3.32T + 31T^{2} \)
37 \( 1 + (-2.44 + 2.44i)T - 37iT^{2} \)
41 \( 1 + 0.648iT - 41T^{2} \)
43 \( 1 + (-0.819 - 0.819i)T + 43iT^{2} \)
47 \( 1 + (6.28 - 6.28i)T - 47iT^{2} \)
53 \( 1 + (-5.60 + 5.60i)T - 53iT^{2} \)
59 \( 1 + 6.12iT - 59T^{2} \)
61 \( 1 + 5.13iT - 61T^{2} \)
67 \( 1 + (-4.90 + 4.90i)T - 67iT^{2} \)
71 \( 1 + 4.13iT - 71T^{2} \)
73 \( 1 + (-4.69 + 4.69i)T - 73iT^{2} \)
79 \( 1 - 1.10iT - 79T^{2} \)
83 \( 1 + (6.27 - 6.27i)T - 83iT^{2} \)
89 \( 1 - 15.3T + 89T^{2} \)
97 \( 1 + (5.42 + 5.42i)T + 97iT^{2} \)
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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

−13.31392036058701190654395566067, −12.07508458563454997051930812054, −11.26582485455751662428462827409, −10.78669510074753323085122304271, −9.458529760889541553965056848449, −8.219367999855435283993312197260, −6.75110967973632144651317918978, −6.36841632920498996939422343527, −3.95848012795265708272927494051, −1.84166702701052026193683283414, 1.03329230054131953142505625594, 4.31734633457101120041564518952, 5.70261692967966253596328939724, 6.65857148542271416309567901990, 8.248539626621148496198586290185, 9.082824754313948646960085702721, 10.25636011620898010477701897720, 11.14412061025996702036605104310, 12.07755134781857731913954440932, 13.23644209696087993552627261176

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