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Properties

Label 2-120-120.53-c3-0-2
Degree $2$
Conductor $120$
Sign $-0.534 + 0.845i$
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  + (1.04 + 2.62i)2-s + (−5.18 + 0.323i)3-s + (−5.82 + 5.47i)4-s + (4.01 + 10.4i)5-s + (−6.25 − 13.3i)6-s + (−2.52 + 2.52i)7-s + (−20.4 − 9.62i)8-s + (26.7 − 3.35i)9-s + (−23.2 + 21.4i)10-s − 64.3·11-s + (28.4 − 30.2i)12-s + (30.6 − 30.6i)13-s + (−9.26 − 4.00i)14-s + (−24.1 − 52.8i)15-s + (3.97 − 63.8i)16-s + (−69.0 − 69.0i)17-s + ⋯
L(s)  = 1  + (0.368 + 0.929i)2-s + (−0.998 + 0.0621i)3-s + (−0.728 + 0.684i)4-s + (0.358 + 0.933i)5-s + (−0.425 − 0.905i)6-s + (−0.136 + 0.136i)7-s + (−0.905 − 0.425i)8-s + (0.992 − 0.124i)9-s + (−0.735 + 0.677i)10-s − 1.76·11-s + (0.684 − 0.728i)12-s + (0.653 − 0.653i)13-s + (−0.176 − 0.0765i)14-s + (−0.416 − 0.909i)15-s + (0.0621 − 0.998i)16-s + (−0.985 − 0.985i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.534 + 0.845i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.534 + 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(120\)    =    \(2^{3} \cdot 3 \cdot 5\)
Sign: $-0.534 + 0.845i$
Analytic conductor: \(7.08022\)
Root analytic conductor: \(2.66087\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{120} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 120,\ (\ :3/2),\ -0.534 + 0.845i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.215792 - 0.391683i\)
\(L(\frac12)\) \(\approx\) \(0.215792 - 0.391683i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.04 - 2.62i)T \)
3 \( 1 + (5.18 - 0.323i)T \)
5 \( 1 + (-4.01 - 10.4i)T \)
good7 \( 1 + (2.52 - 2.52i)T - 343iT^{2} \)
11 \( 1 + 64.3T + 1.33e3T^{2} \)
13 \( 1 + (-30.6 + 30.6i)T - 2.19e3iT^{2} \)
17 \( 1 + (69.0 + 69.0i)T + 4.91e3iT^{2} \)
19 \( 1 - 70.7T + 6.85e3T^{2} \)
23 \( 1 + (60.8 - 60.8i)T - 1.21e4iT^{2} \)
29 \( 1 - 184. iT - 2.43e4T^{2} \)
31 \( 1 + 108.T + 2.97e4T^{2} \)
37 \( 1 + (-91.3 - 91.3i)T + 5.06e4iT^{2} \)
41 \( 1 - 244. iT - 6.89e4T^{2} \)
43 \( 1 + (90.0 - 90.0i)T - 7.95e4iT^{2} \)
47 \( 1 + (271. + 271. i)T + 1.03e5iT^{2} \)
53 \( 1 + (368. + 368. i)T + 1.48e5iT^{2} \)
59 \( 1 - 608. iT - 2.05e5T^{2} \)
61 \( 1 + 13.7iT - 2.26e5T^{2} \)
67 \( 1 + (-111. - 111. i)T + 3.00e5iT^{2} \)
71 \( 1 - 501. iT - 3.57e5T^{2} \)
73 \( 1 + (167. + 167. i)T + 3.89e5iT^{2} \)
79 \( 1 + 670. iT - 4.93e5T^{2} \)
83 \( 1 + (278. + 278. i)T + 5.71e5iT^{2} \)
89 \( 1 - 1.18e3T + 7.04e5T^{2} \)
97 \( 1 + (669. - 669. i)T - 9.12e5iT^{2} \)
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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

−13.47543302918562993313924540061, −13.04388910892515192073949440769, −11.60451291069343736611698772591, −10.60978913783603372932320067900, −9.570163822539634678773656662153, −7.85292257184980655850522216546, −6.92535052056201955755311784440, −5.81448304896439952240425374112, −5.00653949595152726328212486392, −3.11474554925611721148974268671, 0.22877915783993799355595367056, 1.93009477484946789136489063936, 4.18945231588844821456965533974, 5.23560076497897467025530377339, 6.17929893698110175593337847532, 8.147784581784055256036171427805, 9.506539593002624556013704364631, 10.47419953405602249759564275401, 11.27942504933858170253823381628, 12.43208870370573376166782864661

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