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Properties

Label 2-120-120.53-c3-0-22
Degree $2$
Conductor $120$
Sign $0.882 + 0.469i$
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  + (−2.82 − 0.0740i)2-s + (−4.92 − 1.65i)3-s + (7.98 + 0.418i)4-s + (9.98 − 5.02i)5-s + (13.8 + 5.03i)6-s + (−17.9 + 17.9i)7-s + (−22.5 − 1.77i)8-s + (21.5 + 16.2i)9-s + (−28.6 + 13.4i)10-s − 22.2·11-s + (−38.6 − 15.2i)12-s + (31.5 − 31.5i)13-s + (51.9 − 49.3i)14-s + (−57.5 + 8.27i)15-s + (63.6 + 6.68i)16-s + (17.9 + 17.9i)17-s + ⋯
L(s)  = 1  + (−0.999 − 0.0261i)2-s + (−0.948 − 0.317i)3-s + (0.998 + 0.0523i)4-s + (0.893 − 0.449i)5-s + (0.939 + 0.342i)6-s + (−0.967 + 0.967i)7-s + (−0.996 − 0.0784i)8-s + (0.798 + 0.602i)9-s + (−0.904 + 0.425i)10-s − 0.609·11-s + (−0.930 − 0.366i)12-s + (0.673 − 0.673i)13-s + (0.992 − 0.941i)14-s + (−0.989 + 0.142i)15-s + (0.994 + 0.104i)16-s + (0.255 + 0.255i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.882 + 0.469i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.882 + 0.469i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(120\)    =    \(2^{3} \cdot 3 \cdot 5\)
Sign: $0.882 + 0.469i$
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: \((2,\ 120,\ (\ :3/2),\ 0.882 + 0.469i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.777751 - 0.194148i\)
\(L(\frac12)\) \(\approx\) \(0.777751 - 0.194148i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.82 + 0.0740i)T \)
3 \( 1 + (4.92 + 1.65i)T \)
5 \( 1 + (-9.98 + 5.02i)T \)
good7 \( 1 + (17.9 - 17.9i)T - 343iT^{2} \)
11 \( 1 + 22.2T + 1.33e3T^{2} \)
13 \( 1 + (-31.5 + 31.5i)T - 2.19e3iT^{2} \)
17 \( 1 + (-17.9 - 17.9i)T + 4.91e3iT^{2} \)
19 \( 1 - 149.T + 6.85e3T^{2} \)
23 \( 1 + (-44.6 + 44.6i)T - 1.21e4iT^{2} \)
29 \( 1 + 145. iT - 2.43e4T^{2} \)
31 \( 1 - 102.T + 2.97e4T^{2} \)
37 \( 1 + (-2.90 - 2.90i)T + 5.06e4iT^{2} \)
41 \( 1 + 163. iT - 6.89e4T^{2} \)
43 \( 1 + (-251. + 251. i)T - 7.95e4iT^{2} \)
47 \( 1 + (-269. - 269. i)T + 1.03e5iT^{2} \)
53 \( 1 + (-411. - 411. i)T + 1.48e5iT^{2} \)
59 \( 1 + 91.0iT - 2.05e5T^{2} \)
61 \( 1 - 602. iT - 2.26e5T^{2} \)
67 \( 1 + (571. + 571. i)T + 3.00e5iT^{2} \)
71 \( 1 - 802. iT - 3.57e5T^{2} \)
73 \( 1 + (-348. - 348. i)T + 3.89e5iT^{2} \)
79 \( 1 + 923. iT - 4.93e5T^{2} \)
83 \( 1 + (501. + 501. i)T + 5.71e5iT^{2} \)
89 \( 1 + 178.T + 7.04e5T^{2} \)
97 \( 1 + (82.1 - 82.1i)T - 9.12e5iT^{2} \)
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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

−12.62056383603708346127722188842, −11.91833781574543455206229966855, −10.58537404779568957370263785012, −9.827834564367489941250252943530, −8.827125660357668945137898236467, −7.45129559398175102098756576485, −6.03778993174659842786933682298, −5.58637129313186704313401665445, −2.65576780959218174488410195866, −0.885189319556432306781799910127, 1.04296968292396136023994196959, 3.28415816451694882838693650965, 5.49034475729711459923900213021, 6.59763545416809245685090915059, 7.32921804724710367404631299494, 9.318156618231400206499904181299, 9.959194569942108880227522698009, 10.71703284602521542850939067199, 11.63771492675840689153463538889, 12.99713693109497052953720667639

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