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Properties

Label 2-120-120.77-c1-0-17
Degree $2$
Conductor $120$
Sign $-0.162 + 0.986i$
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  + (0.0941 − 1.41i)2-s + (1.68 − 0.416i)3-s + (−1.98 − 0.265i)4-s + (−1.62 − 1.54i)5-s + (−0.429 − 2.41i)6-s + (−0.361 − 0.361i)7-s + (−0.561 + 2.77i)8-s + (2.65 − 1.40i)9-s + (−2.32 + 2.14i)10-s + 2.63·11-s + (−3.44 + 0.378i)12-s + (3.49 + 3.49i)13-s + (−0.544 + 0.476i)14-s + (−3.36 − 1.91i)15-s + (3.85 + 1.05i)16-s + (−3.61 + 3.61i)17-s + ⋯
L(s)  = 1  + (0.0665 − 0.997i)2-s + (0.970 − 0.240i)3-s + (−0.991 − 0.132i)4-s + (−0.724 − 0.688i)5-s + (−0.175 − 0.984i)6-s + (−0.136 − 0.136i)7-s + (−0.198 + 0.980i)8-s + (0.884 − 0.466i)9-s + (−0.735 + 0.677i)10-s + 0.794·11-s + (−0.994 + 0.109i)12-s + (0.968 + 0.968i)13-s + (−0.145 + 0.127i)14-s + (−0.869 − 0.494i)15-s + (0.964 + 0.263i)16-s + (−0.876 + 0.876i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.162 + 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.162 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(120\)    =    \(2^{3} \cdot 3 \cdot 5\)
Sign: $-0.162 + 0.986i$
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: \((2,\ 120,\ (\ :1/2),\ -0.162 + 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.788405 - 0.928964i\)
\(L(\frac12)\) \(\approx\) \(0.788405 - 0.928964i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0941 + 1.41i)T \)
3 \( 1 + (-1.68 + 0.416i)T \)
5 \( 1 + (1.62 + 1.54i)T \)
good7 \( 1 + (0.361 + 0.361i)T + 7iT^{2} \)
11 \( 1 - 2.63T + 11T^{2} \)
13 \( 1 + (-3.49 - 3.49i)T + 13iT^{2} \)
17 \( 1 + (3.61 - 3.61i)T - 17iT^{2} \)
19 \( 1 + 0.672T + 19T^{2} \)
23 \( 1 + (4.31 + 4.31i)T + 23iT^{2} \)
29 \( 1 - 4.76iT - 29T^{2} \)
31 \( 1 - 3.73T + 31T^{2} \)
37 \( 1 + (2.82 - 2.82i)T - 37iT^{2} \)
41 \( 1 - 4.10iT - 41T^{2} \)
43 \( 1 + (7.57 + 7.57i)T + 43iT^{2} \)
47 \( 1 + (0.987 - 0.987i)T - 47iT^{2} \)
53 \( 1 + (-0.646 + 0.646i)T - 53iT^{2} \)
59 \( 1 + 4.92iT - 59T^{2} \)
61 \( 1 + 6.07iT - 61T^{2} \)
67 \( 1 + (0.349 - 0.349i)T - 67iT^{2} \)
71 \( 1 - 8.63iT - 71T^{2} \)
73 \( 1 + (11.3 - 11.3i)T - 73iT^{2} \)
79 \( 1 + 4.07iT - 79T^{2} \)
83 \( 1 + (8.53 - 8.53i)T - 83iT^{2} \)
89 \( 1 - 6.58T + 89T^{2} \)
97 \( 1 + (0.660 + 0.660i)T + 97iT^{2} \)
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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

−13.11053456099501495240386447533, −12.24364492295058865531074571760, −11.32017411827755297636202114051, −9.988307063047216566881827356475, −8.685685498238562889928361270647, −8.494533352125662097183395391996, −6.63913437022979664046088284926, −4.39966148601663446658278058449, −3.65142442294130293079231987175, −1.65872579289959148908992274036, 3.28077003028824354403055721232, 4.34727928756796896821459393332, 6.20189654890794432174054463920, 7.37262828865151667241407651760, 8.249042944439835530037007531518, 9.216430376690938080395120366016, 10.36443949711949268437837265310, 11.80653016456000233000223332048, 13.25281061706049759668551619614, 13.93407635008635524530150742146

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