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Properties

Label 2-120-120.59-c3-0-30
Degree $2$
Conductor $120$
Sign $0.819 + 0.573i$
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  + (−2.73 + 0.732i)2-s + (−2.64 − 4.47i)3-s + (6.92 − 4.00i)4-s + (10.4 + 3.92i)5-s + (10.4 + 10.2i)6-s + 4.10·7-s + (−15.9 + 16.0i)8-s + (−13.0 + 23.6i)9-s + (−31.4 − 3.06i)10-s − 5.96i·11-s + (−36.2 − 20.4i)12-s + 33.0·13-s + (−11.2 + 3.00i)14-s + (−10.0 − 57.2i)15-s + (31.9 − 55.4i)16-s + 50.6·17-s + ⋯
L(s)  = 1  + (−0.965 + 0.259i)2-s + (−0.508 − 0.861i)3-s + (0.865 − 0.500i)4-s + (0.936 + 0.351i)5-s + (0.714 + 0.699i)6-s + 0.221·7-s + (−0.706 + 0.707i)8-s + (−0.483 + 0.875i)9-s + (−0.995 − 0.0968i)10-s − 0.163i·11-s + (−0.871 − 0.491i)12-s + 0.705·13-s + (−0.213 + 0.0573i)14-s + (−0.173 − 0.984i)15-s + (0.498 − 0.866i)16-s + 0.722·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.819 + 0.573i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.819 + 0.573i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(120\)    =    \(2^{3} \cdot 3 \cdot 5\)
Sign: $0.819 + 0.573i$
Analytic conductor: \(7.08022\)
Root analytic conductor: \(2.66087\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{120} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 120,\ (\ :3/2),\ 0.819 + 0.573i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.03344 - 0.325511i\)
\(L(\frac12)\) \(\approx\) \(1.03344 - 0.325511i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.73 - 0.732i)T \)
3 \( 1 + (2.64 + 4.47i)T \)
5 \( 1 + (-10.4 - 3.92i)T \)
good7 \( 1 - 4.10T + 343T^{2} \)
11 \( 1 + 5.96iT - 1.33e3T^{2} \)
13 \( 1 - 33.0T + 2.19e3T^{2} \)
17 \( 1 - 50.6T + 4.91e3T^{2} \)
19 \( 1 - 74.1T + 6.85e3T^{2} \)
23 \( 1 + 184. iT - 1.21e4T^{2} \)
29 \( 1 + 98.3T + 2.43e4T^{2} \)
31 \( 1 + 192. iT - 2.97e4T^{2} \)
37 \( 1 - 350.T + 5.06e4T^{2} \)
41 \( 1 + 292. iT - 6.89e4T^{2} \)
43 \( 1 - 80.8iT - 7.95e4T^{2} \)
47 \( 1 + 63.1iT - 1.03e5T^{2} \)
53 \( 1 + 178. iT - 1.48e5T^{2} \)
59 \( 1 - 479. iT - 2.05e5T^{2} \)
61 \( 1 - 635. iT - 2.26e5T^{2} \)
67 \( 1 - 288. iT - 3.00e5T^{2} \)
71 \( 1 - 1.06e3T + 3.57e5T^{2} \)
73 \( 1 - 980. iT - 3.89e5T^{2} \)
79 \( 1 + 804. iT - 4.93e5T^{2} \)
83 \( 1 + 300.T + 5.71e5T^{2} \)
89 \( 1 - 1.11e3iT - 7.04e5T^{2} \)
97 \( 1 + 1.21e3iT - 9.12e5T^{2} \)
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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

−12.86133507144015835753011783058, −11.59054176470156419456473369895, −10.78441107859683661267410107251, −9.759452689137890556765908111473, −8.487154224067568933786365349409, −7.40749976938781413663367423877, −6.31215640547608428652997393740, −5.53472971034587753049075188206, −2.47408219385506604658810368051, −1.02980194465050749378002030977, 1.29386079228892506296343253379, 3.36130179326631872327721562395, 5.25227002745845170794244817513, 6.33901104706569913996485301887, 7.949601423614781140633791711954, 9.332743087569820196407024533259, 9.722797264566070273995417260856, 10.87468671666714076527953951681, 11.68214318980568211859748274218, 12.84938008804796850199110568112

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