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Properties

Label 2-120-120.59-c3-0-18
Degree $2$
Conductor $120$
Sign $0.983 - 0.181i$
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  + (−1.65 + 2.29i)2-s + (−4.82 − 1.93i)3-s + (−2.51 − 7.59i)4-s + (−7.20 + 8.54i)5-s + (12.4 − 7.85i)6-s − 30.0·7-s + (21.5 + 6.82i)8-s + (19.5 + 18.6i)9-s + (−7.65 − 30.6i)10-s − 35.6i·11-s + (−2.55 + 41.4i)12-s + 58.6·13-s + (49.7 − 68.8i)14-s + (51.2 − 27.3i)15-s + (−51.3 + 38.1i)16-s − 14.3·17-s + ⋯
L(s)  = 1  + (−0.585 + 0.810i)2-s + (−0.928 − 0.371i)3-s + (−0.313 − 0.949i)4-s + (−0.644 + 0.764i)5-s + (0.845 − 0.534i)6-s − 1.62·7-s + (0.953 + 0.301i)8-s + (0.723 + 0.690i)9-s + (−0.242 − 0.970i)10-s − 0.976i·11-s + (−0.0615 + 0.998i)12-s + 1.25·13-s + (0.950 − 1.31i)14-s + (0.882 − 0.470i)15-s + (−0.803 + 0.595i)16-s − 0.205·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.181i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.983 - 0.181i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(120\)    =    \(2^{3} \cdot 3 \cdot 5\)
Sign: $0.983 - 0.181i$
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: \((2,\ 120,\ (\ :3/2),\ 0.983 - 0.181i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.528058 + 0.0484124i\)
\(L(\frac12)\) \(\approx\) \(0.528058 + 0.0484124i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.65 - 2.29i)T \)
3 \( 1 + (4.82 + 1.93i)T \)
5 \( 1 + (7.20 - 8.54i)T \)
good7 \( 1 + 30.0T + 343T^{2} \)
11 \( 1 + 35.6iT - 1.33e3T^{2} \)
13 \( 1 - 58.6T + 2.19e3T^{2} \)
17 \( 1 + 14.3T + 4.91e3T^{2} \)
19 \( 1 - 106.T + 6.85e3T^{2} \)
23 \( 1 - 47.7iT - 1.21e4T^{2} \)
29 \( 1 - 49.5T + 2.43e4T^{2} \)
31 \( 1 - 181. iT - 2.97e4T^{2} \)
37 \( 1 - 113.T + 5.06e4T^{2} \)
41 \( 1 - 53.6iT - 6.89e4T^{2} \)
43 \( 1 + 490. iT - 7.95e4T^{2} \)
47 \( 1 + 441. iT - 1.03e5T^{2} \)
53 \( 1 + 65.6iT - 1.48e5T^{2} \)
59 \( 1 + 406. iT - 2.05e5T^{2} \)
61 \( 1 - 213. iT - 2.26e5T^{2} \)
67 \( 1 - 825. iT - 3.00e5T^{2} \)
71 \( 1 - 157.T + 3.57e5T^{2} \)
73 \( 1 - 242. iT - 3.89e5T^{2} \)
79 \( 1 - 107. iT - 4.93e5T^{2} \)
83 \( 1 - 1.15e3T + 5.71e5T^{2} \)
89 \( 1 + 750. iT - 7.04e5T^{2} \)
97 \( 1 + 83.2iT - 9.12e5T^{2} \)
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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

−13.24149488446692525184115346236, −11.78162359550188760334390324906, −10.82209424358109925379038392889, −9.972763876660120861080732307689, −8.590285000359589609247597982174, −7.22593381605970798868260280741, −6.49268083037701251169662602749, −5.64231614543152652089393671086, −3.54487127250043970902043184477, −0.59675902356634427387737661191, 0.849518751891917636464977430070, 3.44572051937694015522049235705, 4.54231963217077823280862609900, 6.28957770951676946223575063374, 7.62827230000392002197237700592, 9.223594630287705607433450124150, 9.737015455660715415267895532556, 10.93147736585823441157586337263, 11.90838279775828522299163628135, 12.67027076261770106514884958124

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