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Properties

Label 2-120-120.77-c1-0-8
Degree $2$
Conductor $120$
Sign $0.311 - 0.950i$
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  + (0.533 + 1.30i)2-s + (1.59 + 0.667i)3-s + (−1.43 + 1.39i)4-s + (0.143 − 2.23i)5-s + (−0.0218 + 2.44i)6-s + (0.582 + 0.582i)7-s + (−2.59 − 1.13i)8-s + (2.10 + 2.13i)9-s + (2.99 − 1.00i)10-s − 3.68·11-s + (−3.22 + 1.27i)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  + (0.377 + 0.926i)2-s + (0.922 + 0.385i)3-s + (−0.715 + 0.698i)4-s + (0.0639 − 0.997i)5-s + (−0.00892 + 0.999i)6-s + (0.220 + 0.220i)7-s + (−0.916 − 0.399i)8-s + (0.703 + 0.711i)9-s + (0.948 − 0.316i)10-s − 1.11·11-s + (−0.929 + 0.368i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.311 - 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.311 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(120\)    =    \(2^{3} \cdot 3 \cdot 5\)
Sign: $0.311 - 0.950i$
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: \((2,\ 120,\ (\ :1/2),\ 0.311 - 0.950i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.18738 + 0.860090i\)
\(L(\frac12)\) \(\approx\) \(1.18738 + 0.860090i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.533 - 1.30i)T \)
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

−13.64813866771310253925024524517, −13.07465919548619245949263750474, −12.08267538006752368062601360812, −10.11774493687689452953203006867, −9.209465464448066445009749897079, −8.099143745639525942010466027594, −7.55028747711155399544709683977, −5.42216905625436586800682869813, −4.75019337651072685324656060181, −3.01911731985750245350868598566, 2.22467802234956333354550324128, 3.28466127136667913066239566178, 4.89356260491190917983625212336, 6.76419059555456871234573654010, 7.85855487078356810439648631930, 9.367623988962739086909988371636, 10.12445798610275017196168575639, 11.28000860943481382103855553457, 12.28420714829393539449789436494, 13.43137421447527098548799335588

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