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Properties

Label 2-60-12.11-c3-0-20
Degree $2$
Conductor $60$
Sign $-0.227 + 0.973i$
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  + (0.235 − 2.81i)2-s + (5.18 + 0.327i)3-s + (−7.88 − 1.32i)4-s − 5i·5-s + (2.14 − 14.5i)6-s − 26.3i·7-s + (−5.59 + 21.9i)8-s + (26.7 + 3.40i)9-s + (−14.0 − 1.17i)10-s + 41.8·11-s + (−40.4 − 9.46i)12-s − 62.0·13-s + (−74.1 − 6.19i)14-s + (1.63 − 25.9i)15-s + (60.4 + 20.9i)16-s + 57.4i·17-s + ⋯
L(s)  = 1  + (0.0832 − 0.996i)2-s + (0.998 + 0.0631i)3-s + (−0.986 − 0.165i)4-s − 0.447i·5-s + (0.145 − 0.989i)6-s − 1.42i·7-s + (−0.247 + 0.968i)8-s + (0.992 + 0.125i)9-s + (−0.445 − 0.0372i)10-s + 1.14·11-s + (−0.973 − 0.227i)12-s − 1.32·13-s + (−1.41 − 0.118i)14-s + (0.0282 − 0.446i)15-s + (0.944 + 0.327i)16-s + 0.819i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.227 + 0.973i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.227 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $-0.227 + 0.973i$
Analytic conductor: \(3.54011\)
Root analytic conductor: \(1.88151\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :3/2),\ -0.227 + 0.973i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.11927 - 1.41135i\)
\(L(\frac12)\) \(\approx\) \(1.11927 - 1.41135i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.235 + 2.81i)T \)
3 \( 1 + (-5.18 - 0.327i)T \)
5 \( 1 + 5iT \)
good7 \( 1 + 26.3iT - 343T^{2} \)
11 \( 1 - 41.8T + 1.33e3T^{2} \)
13 \( 1 + 62.0T + 2.19e3T^{2} \)
17 \( 1 - 57.4iT - 4.91e3T^{2} \)
19 \( 1 - 72.8iT - 6.85e3T^{2} \)
23 \( 1 - 144.T + 1.21e4T^{2} \)
29 \( 1 - 100. iT - 2.43e4T^{2} \)
31 \( 1 - 130. iT - 2.97e4T^{2} \)
37 \( 1 - 10.8T + 5.06e4T^{2} \)
41 \( 1 - 17.9iT - 6.89e4T^{2} \)
43 \( 1 + 366. iT - 7.95e4T^{2} \)
47 \( 1 + 257.T + 1.03e5T^{2} \)
53 \( 1 + 147. iT - 1.48e5T^{2} \)
59 \( 1 + 442.T + 2.05e5T^{2} \)
61 \( 1 + 279.T + 2.26e5T^{2} \)
67 \( 1 - 735. iT - 3.00e5T^{2} \)
71 \( 1 - 299.T + 3.57e5T^{2} \)
73 \( 1 + 740.T + 3.89e5T^{2} \)
79 \( 1 + 209. iT - 4.93e5T^{2} \)
83 \( 1 + 89.4T + 5.71e5T^{2} \)
89 \( 1 + 353. iT - 7.04e5T^{2} \)
97 \( 1 - 902.T + 9.12e5T^{2} \)
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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

−14.18781782182787041549214266965, −13.11933489421077683771959977046, −12.20958458620101675262472185349, −10.60860612179586479312946602239, −9.720379751280999912148831843356, −8.627586757632545580886542333437, −7.23278414281926625833326200261, −4.60600795293101572462472904453, −3.52550622714176257607132043884, −1.42225602560883780247777392121, 2.83455565733717454936709864289, 4.78572818970331037785019942308, 6.52550765448788756023158247395, 7.63513117154542536900777595940, 9.073067448394817080573391508918, 9.510038384720564601786740657099, 11.82810575692700129290685104845, 12.97508832768643567958061459590, 14.21704067644159090188437363432, 14.95806019206869921013031470657

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