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Properties

Label 2-60-15.2-c5-0-4
Degree $2$
Conductor $60$
Sign $0.743 + 0.668i$
Analytic cond. $9.62302$
Root an. cond. $3.10210$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−15.0 − 4.14i)3-s + (29.4 + 47.5i)5-s + (−98.6 − 98.6i)7-s + (208. + 124. i)9-s + 208. i·11-s + (823. − 823. i)13-s + (−245. − 836. i)15-s + (1.33e3 − 1.33e3i)17-s − 2.16e3i·19-s + (1.07e3 + 1.89e3i)21-s + (2.06e3 + 2.06e3i)23-s + (−1.39e3 + 2.79e3i)25-s + (−2.61e3 − 2.73e3i)27-s − 43.7·29-s + 1.05e3·31-s + ⋯
L(s)  = 1  + (−0.963 − 0.265i)3-s + (0.526 + 0.850i)5-s + (−0.760 − 0.760i)7-s + (0.858 + 0.512i)9-s + 0.518i·11-s + (1.35 − 1.35i)13-s + (−0.281 − 0.959i)15-s + (1.12 − 1.12i)17-s − 1.37i·19-s + (0.530 + 0.935i)21-s + (0.812 + 0.812i)23-s + (−0.446 + 0.894i)25-s + (−0.691 − 0.722i)27-s − 0.00965·29-s + 0.197·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.743 + 0.668i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.743 + 0.668i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.743 + 0.668i$
Analytic conductor: \(9.62302\)
Root analytic conductor: \(3.10210\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :5/2),\ 0.743 + 0.668i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.19324 - 0.457469i\)
\(L(\frac12)\) \(\approx\) \(1.19324 - 0.457469i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (15.0 + 4.14i)T \)
5 \( 1 + (-29.4 - 47.5i)T \)
good7 \( 1 + (98.6 + 98.6i)T + 1.68e4iT^{2} \)
11 \( 1 - 208. iT - 1.61e5T^{2} \)
13 \( 1 + (-823. + 823. i)T - 3.71e5iT^{2} \)
17 \( 1 + (-1.33e3 + 1.33e3i)T - 1.41e6iT^{2} \)
19 \( 1 + 2.16e3iT - 2.47e6T^{2} \)
23 \( 1 + (-2.06e3 - 2.06e3i)T + 6.43e6iT^{2} \)
29 \( 1 + 43.7T + 2.05e7T^{2} \)
31 \( 1 - 1.05e3T + 2.86e7T^{2} \)
37 \( 1 + (-2.19e3 - 2.19e3i)T + 6.93e7iT^{2} \)
41 \( 1 - 7.32e3iT - 1.15e8T^{2} \)
43 \( 1 + (5.65e3 - 5.65e3i)T - 1.47e8iT^{2} \)
47 \( 1 + (-1.70e4 + 1.70e4i)T - 2.29e8iT^{2} \)
53 \( 1 + (1.61e4 + 1.61e4i)T + 4.18e8iT^{2} \)
59 \( 1 - 1.34e4T + 7.14e8T^{2} \)
61 \( 1 + 1.89e3T + 8.44e8T^{2} \)
67 \( 1 + (-8.17e3 - 8.17e3i)T + 1.35e9iT^{2} \)
71 \( 1 - 1.00e4iT - 1.80e9T^{2} \)
73 \( 1 + (-2.44e4 + 2.44e4i)T - 2.07e9iT^{2} \)
79 \( 1 + 8.56e4iT - 3.07e9T^{2} \)
83 \( 1 + (9.10e3 + 9.10e3i)T + 3.93e9iT^{2} \)
89 \( 1 + 3.42e4T + 5.58e9T^{2} \)
97 \( 1 + (997. + 997. i)T + 8.58e9iT^{2} \)
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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

−13.55858976311862516531721377423, −13.07930053118372626130195564362, −11.49275433858153508909390939514, −10.55560332814855095896968088282, −9.702395196565004690599913903291, −7.46922339167595625244653000827, −6.57281824065611116125843253318, −5.32297508140803461229483819187, −3.22099672712371521519192306871, −0.830441569371714267422289697272, 1.30384002838482245781207267966, 3.95271242376315976845152286758, 5.69314025907126527736744340491, 6.29302302106393600943518463450, 8.558529421031779756147604342650, 9.574706281514976790919064194632, 10.78480615033209125987312168646, 12.15022381264366259418970604375, 12.76484509541826023344332328784, 14.11954074729425324889626329483

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