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Properties

Label 2-60-60.59-c3-0-8
Degree $2$
Conductor $60$
Sign $0.952 - 0.304i$
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  − 2.82i·2-s + (1.58 + 4.94i)3-s − 8.00·4-s + 11.1i·5-s + (14.0 − 4.47i)6-s + 34.7·7-s + 22.6i·8-s + (−22 + 15.6i)9-s + 31.6·10-s + (−12.6 − 39.5i)12-s − 98.3i·14-s + (−55.3 + 17.6i)15-s + 64.0·16-s + (44.2 + 62.2i)18-s − 89.4i·20-s + (55.0 + 172. i)21-s + ⋯
L(s)  = 1  − 0.999i·2-s + (0.304 + 0.952i)3-s − 1.00·4-s + 0.999i·5-s + (0.952 − 0.304i)6-s + 1.87·7-s + 1.00i·8-s + (−0.814 + 0.579i)9-s + 1.00·10-s + (−0.304 − 0.952i)12-s − 1.87i·14-s + (−0.952 + 0.304i)15-s + 1.00·16-s + (0.579 + 0.814i)18-s − 1.00i·20-s + (0.571 + 1.78i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 - 0.304i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.952 - 0.304i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.952 - 0.304i$
Analytic conductor: \(3.54011\)
Root analytic conductor: \(1.88151\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :3/2),\ 0.952 - 0.304i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.52841 + 0.238188i\)
\(L(\frac12)\) \(\approx\) \(1.52841 + 0.238188i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 2.82iT \)
3 \( 1 + (-1.58 - 4.94i)T \)
5 \( 1 - 11.1iT \)
good7 \( 1 - 34.7T + 343T^{2} \)
11 \( 1 + 1.33e3T^{2} \)
13 \( 1 - 2.19e3T^{2} \)
17 \( 1 + 4.91e3T^{2} \)
19 \( 1 - 6.85e3T^{2} \)
23 \( 1 + 94.7iT - 1.21e4T^{2} \)
29 \( 1 - 62.6iT - 2.43e4T^{2} \)
31 \( 1 - 2.97e4T^{2} \)
37 \( 1 - 5.06e4T^{2} \)
41 \( 1 + 460. iT - 6.89e4T^{2} \)
43 \( 1 - 376.T + 7.95e4T^{2} \)
47 \( 1 + 425. iT - 1.03e5T^{2} \)
53 \( 1 + 1.48e5T^{2} \)
59 \( 1 + 2.05e5T^{2} \)
61 \( 1 + 952T + 2.26e5T^{2} \)
67 \( 1 - 774.T + 3.00e5T^{2} \)
71 \( 1 + 3.57e5T^{2} \)
73 \( 1 - 3.89e5T^{2} \)
79 \( 1 - 4.93e5T^{2} \)
83 \( 1 - 108. iT - 5.71e5T^{2} \)
89 \( 1 - 948. iT - 7.04e5T^{2} \)
97 \( 1 - 9.12e5T^{2} \)
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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

−14.43215365198126514997577301282, −13.91779463531260911499388811800, −11.94532562022944738302220995765, −10.94045150096948418940245322277, −10.47409595159751539995875544676, −8.958939443464264464719319729533, −7.86107133459467140753643875731, −5.25832508906662276235860247802, −4.00492524771816173921710435288, −2.31228669392935700702097668963, 1.30835252347262208714899734917, 4.57245708444771568703551669400, 5.79466902860976828786439429536, 7.60362716723354735241882698995, 8.188437353256050882289912562211, 9.235938754708454756838525140473, 11.42602792739656245107101292246, 12.57207000025252761453774250254, 13.67139930232028788883033667445, 14.43103434247992646642994134116

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