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Properties

Label 2-60-5.3-c4-0-2
Degree $2$
Conductor $60$
Sign $0.974 + 0.222i$
Analytic cond. $6.20219$
Root an. cond. $2.49042$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.67 + 3.67i)3-s + (11.0 − 22.4i)5-s + (21.1 − 21.1i)7-s + 27i·9-s + 160.·11-s + (70.6 + 70.6i)13-s + (122. − 41.9i)15-s + (27.2 − 27.2i)17-s − 334. i·19-s + 155.·21-s + (259. + 259. i)23-s + (−381. − 494. i)25-s + (−99.2 + 99.2i)27-s + 884. i·29-s − 1.88e3·31-s + ⋯
L(s)  = 1  + (0.408 + 0.408i)3-s + (0.440 − 0.897i)5-s + (0.432 − 0.432i)7-s + 0.333i·9-s + 1.32·11-s + (0.418 + 0.418i)13-s + (0.546 − 0.186i)15-s + (0.0943 − 0.0943i)17-s − 0.927i·19-s + 0.352·21-s + (0.490 + 0.490i)23-s + (−0.611 − 0.791i)25-s + (−0.136 + 0.136i)27-s + 1.05i·29-s − 1.96·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 + 0.222i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.974 + 0.222i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.974 + 0.222i$
Analytic conductor: \(6.20219\)
Root analytic conductor: \(2.49042\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :2),\ 0.974 + 0.222i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.04338 - 0.230687i\)
\(L(\frac12)\) \(\approx\) \(2.04338 - 0.230687i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-3.67 - 3.67i)T \)
5 \( 1 + (-11.0 + 22.4i)T \)
good7 \( 1 + (-21.1 + 21.1i)T - 2.40e3iT^{2} \)
11 \( 1 - 160.T + 1.46e4T^{2} \)
13 \( 1 + (-70.6 - 70.6i)T + 2.85e4iT^{2} \)
17 \( 1 + (-27.2 + 27.2i)T - 8.35e4iT^{2} \)
19 \( 1 + 334. iT - 1.30e5T^{2} \)
23 \( 1 + (-259. - 259. i)T + 2.79e5iT^{2} \)
29 \( 1 - 884. iT - 7.07e5T^{2} \)
31 \( 1 + 1.88e3T + 9.23e5T^{2} \)
37 \( 1 + (703. - 703. i)T - 1.87e6iT^{2} \)
41 \( 1 + 1.43e3T + 2.82e6T^{2} \)
43 \( 1 + (1.88e3 + 1.88e3i)T + 3.41e6iT^{2} \)
47 \( 1 + (-1.48e3 + 1.48e3i)T - 4.87e6iT^{2} \)
53 \( 1 + (-3.67e3 - 3.67e3i)T + 7.89e6iT^{2} \)
59 \( 1 - 1.08e3iT - 1.21e7T^{2} \)
61 \( 1 + 3.89e3T + 1.38e7T^{2} \)
67 \( 1 + (-4.72e3 + 4.72e3i)T - 2.01e7iT^{2} \)
71 \( 1 + 4.44e3T + 2.54e7T^{2} \)
73 \( 1 + (-5.15e3 - 5.15e3i)T + 2.83e7iT^{2} \)
79 \( 1 - 5.36e3iT - 3.89e7T^{2} \)
83 \( 1 + (-692. - 692. i)T + 4.74e7iT^{2} \)
89 \( 1 + 5.77e3iT - 6.27e7T^{2} \)
97 \( 1 + (6.54e3 - 6.54e3i)T - 8.85e7iT^{2} \)
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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

−14.17898205650309764378107275455, −13.41204792099187814845789769052, −12.04090146228328073782177120153, −10.81776939361850308112636719730, −9.322315910907982236603286080042, −8.724432389635062035191518152692, −7.00659048668625821814492728213, −5.21239583927455857471810146384, −3.87940239872145145947649045004, −1.47088944716939243370426519827, 1.82165442790926178713052259981, 3.57030510345658695220234257061, 5.83931068604989715082959404815, 7.00924897864933460502759912366, 8.421099771106537367739962106304, 9.648194038962791363611067710891, 11.00797657646271043746137653375, 12.10875340524145580178403585854, 13.42291446315996085424420210848, 14.53201987549916484815465176315

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