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Properties

Label 2-60-20.3-c3-0-13
Degree $2$
Conductor $60$
Sign $0.658 + 0.752i$
Analytic cond. $3.54011$
Root an. cond. $1.88151$
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  + (−1.16 + 2.57i)2-s + (2.12 − 2.12i)3-s + (−5.29 − 6.00i)4-s + (2.61 − 10.8i)5-s + (3 + 7.93i)6-s + (−17.4 − 17.4i)7-s + (21.6 − 6.65i)8-s − 8.99i·9-s + (24.9 + 19.3i)10-s − 24.6i·11-s + (−23.9 − 1.50i)12-s + (51.2 + 51.2i)13-s + (65.3 − 24.7i)14-s + (−17.5 − 28.6i)15-s + (−8 + 63.4i)16-s + (48.7 − 48.7i)17-s + ⋯
L(s)  = 1  + (−0.411 + 0.911i)2-s + (0.408 − 0.408i)3-s + (−0.661 − 0.750i)4-s + (0.233 − 0.972i)5-s + (0.204 + 0.540i)6-s + (−0.943 − 0.943i)7-s + (0.955 − 0.294i)8-s − 0.333i·9-s + (0.790 + 0.613i)10-s − 0.676i·11-s + (−0.576 − 0.0361i)12-s + (1.09 + 1.09i)13-s + (1.24 − 0.471i)14-s + (−0.301 − 0.492i)15-s + (−0.125 + 0.992i)16-s + (0.694 − 0.694i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.658 + 0.752i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.658 + 0.752i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.658 + 0.752i$
Analytic conductor: \(3.54011\)
Root analytic conductor: \(1.88151\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :3/2),\ 0.658 + 0.752i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.00471 - 0.455980i\)
\(L(\frac12)\) \(\approx\) \(1.00471 - 0.455980i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.16 - 2.57i)T \)
3 \( 1 + (-2.12 + 2.12i)T \)
5 \( 1 + (-2.61 + 10.8i)T \)
good7 \( 1 + (17.4 + 17.4i)T + 343iT^{2} \)
11 \( 1 + 24.6iT - 1.33e3T^{2} \)
13 \( 1 + (-51.2 - 51.2i)T + 2.19e3iT^{2} \)
17 \( 1 + (-48.7 + 48.7i)T - 4.91e3iT^{2} \)
19 \( 1 + 100.T + 6.85e3T^{2} \)
23 \( 1 + (-52.5 + 52.5i)T - 1.21e4iT^{2} \)
29 \( 1 - 52.9iT - 2.43e4T^{2} \)
31 \( 1 - 180. iT - 2.97e4T^{2} \)
37 \( 1 + (-43.4 + 43.4i)T - 5.06e4iT^{2} \)
41 \( 1 - 250.T + 6.89e4T^{2} \)
43 \( 1 + (-306. + 306. i)T - 7.95e4iT^{2} \)
47 \( 1 + (-267. - 267. i)T + 1.03e5iT^{2} \)
53 \( 1 + (-201. - 201. i)T + 1.48e5iT^{2} \)
59 \( 1 - 74.8T + 2.05e5T^{2} \)
61 \( 1 + 595.T + 2.26e5T^{2} \)
67 \( 1 + (-613. - 613. i)T + 3.00e5iT^{2} \)
71 \( 1 - 293. iT - 3.57e5T^{2} \)
73 \( 1 + (34.7 + 34.7i)T + 3.89e5iT^{2} \)
79 \( 1 + 382.T + 4.93e5T^{2} \)
83 \( 1 + (-457. + 457. i)T - 5.71e5iT^{2} \)
89 \( 1 + 8.16iT - 7.04e5T^{2} \)
97 \( 1 + (94.7 - 94.7i)T - 9.12e5iT^{2} \)
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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

−14.20393182543941455036135818583, −13.59915605209141635557623249927, −12.65113470669277644847263637883, −10.67601996893846132677498449308, −9.283228919086839432111099588011, −8.572984762287685970359730092155, −7.10928627591838929056217749684, −6.00989010853125439310020428105, −4.13007419511954367453353367617, −0.912275248191338279816892460100, 2.49941423277454605138980479710, 3.68286258731451877545177330690, 6.00053741258414160848130400583, 7.895464589895151685877811731350, 9.201746328211199352718451841280, 10.14636871495857316858234638934, 11.02073735348538534486098660913, 12.52953167502797417037514593596, 13.33098948319674781694284989002, 14.82567159775927753753298209280

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