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Properties

Label 2-60-5.2-c4-0-1
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: $\chi_{60} (37, \cdot )$
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

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

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