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Properties

Label 2-120-120.53-c3-0-0
Degree $2$
Conductor $120$
Sign $-0.951 + 0.309i$
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  + (0.0740 + 2.82i)2-s + (−1.65 − 4.92i)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 + (−1.77 − 22.5i)8-s + (−21.5 + 16.2i)9-s + (14.9 + 27.8i)10-s − 22.2·11-s + (15.2 + 38.6i)12-s + (−31.5 + 31.5i)13-s + (−51.9 − 49.3i)14-s + (−41.2 − 40.9i)15-s + (63.6 − 6.68i)16-s + (−17.9 − 17.9i)17-s + ⋯
L(s)  = 1  + (0.0261 + 0.999i)2-s + (−0.317 − 0.948i)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 + (−0.0784 − 0.996i)8-s + (−0.798 + 0.602i)9-s + (0.472 + 0.881i)10-s − 0.609·11-s + (0.366 + 0.930i)12-s + (−0.673 + 0.673i)13-s + (−0.992 − 0.941i)14-s + (−0.709 − 0.704i)15-s + (0.994 − 0.104i)16-s + (−0.255 − 0.255i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.951 + 0.309i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.951 + 0.309i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(120\)    =    \(2^{3} \cdot 3 \cdot 5\)
Sign: $-0.951 + 0.309i$
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: \((2,\ 120,\ (\ :3/2),\ -0.951 + 0.309i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0218219 - 0.137728i\)
\(L(\frac12)\) \(\approx\) \(0.0218219 - 0.137728i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0740 - 2.82i)T \)
3 \( 1 + (1.65 + 4.92i)T \)
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

−13.33524786802548017527959242373, −12.92532007267023606678357966346, −11.96952178418832333668163320584, −10.07900344026990809031678317709, −9.081206287869046318698498205216, −8.123860243348413006116810812776, −6.60994098328852058477313768541, −6.07470659583534022619503227456, −4.91027101202694575252762806978, −2.33492758845616178927811352363, 0.07041525272835997238158363663, 2.63514056301570546218072094900, 3.90714962666107401822198704553, 5.25071198320303663234652932208, 6.57620232381191044820520305862, 8.576417122880167271345616486043, 9.880283228719871689017236159500, 10.30304751082574623403815375957, 10.91731843360626172369335133034, 12.52445979866450438669062469678

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