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Properties

Label 2-120-120.53-c1-0-2
Degree $2$
Conductor $120$
Sign $-0.246 - 0.969i$
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  + (−1.41 − 0.0941i)2-s + (0.416 + 1.68i)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 + (−2.77 − 0.561i)8-s + (−2.65 + 1.40i)9-s + (2.43 − 2.02i)10-s + 2.63·11-s + (0.378 + 3.44i)12-s + (−3.49 + 3.49i)13-s + (0.544 − 0.476i)14-s + (−3.26 − 2.08i)15-s + (3.85 + 1.05i)16-s + (3.61 + 3.61i)17-s + ⋯
L(s)  = 1  + (−0.997 − 0.0665i)2-s + (0.240 + 0.970i)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 + (−0.980 − 0.198i)8-s + (−0.884 + 0.466i)9-s + (0.769 − 0.638i)10-s + 0.794·11-s + (0.109 + 0.994i)12-s + (−0.968 + 0.968i)13-s + (0.145 − 0.127i)14-s + (−0.842 − 0.538i)15-s + (0.964 + 0.263i)16-s + (0.876 + 0.876i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.246 - 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.246 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(120\)    =    \(2^{3} \cdot 3 \cdot 5\)
Sign: $-0.246 - 0.969i$
Analytic conductor: \(0.958204\)
Root analytic conductor: \(0.978879\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{120} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 120,\ (\ :1/2),\ -0.246 - 0.969i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.387251 + 0.498141i\)
\(L(\frac12)\) \(\approx\) \(0.387251 + 0.498141i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 + 0.0941i)T \)
3 \( 1 + (-0.416 - 1.68i)T \)
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

−14.34360678780914511726600028935, −12.21445941320958123168003102452, −11.47506850509659962639056179081, −10.47297560658996592500510125165, −9.620896344291763283605748000566, −8.625488320212957248207440457502, −7.49397795140186322597517730287, −6.25821606753875046110332018125, −4.20011787946726073988815900667, −2.78755002876772390421098382058, 0.965620455948851374096445281244, 3.08818423879855636304220805831, 5.48994113794303824584628921110, 7.11822383306970411121674410420, 7.68370159013067655949604153459, 8.795208228456685411754524824753, 9.711764431371147060825411557029, 11.30335185492922899725571171089, 12.09231161285637708419789312588, 12.83856093100641506945180748673

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