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Properties

Label 2-60-12.11-c3-0-8
Degree $2$
Conductor $60$
Sign $0.998 + 0.0568i$
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  + (−2.29 + 1.64i)2-s + (−5.00 + 1.38i)3-s + (2.56 − 7.57i)4-s − 5i·5-s + (9.23 − 11.4i)6-s + 0.228i·7-s + (6.61 + 21.6i)8-s + (23.1 − 13.8i)9-s + (8.24 + 11.4i)10-s + 57.2·11-s + (−2.36 + 41.5i)12-s + 21.7·13-s + (−0.377 − 0.526i)14-s + (6.90 + 25.0i)15-s + (−50.8 − 38.8i)16-s − 57.8i·17-s + ⋯
L(s)  = 1  + (−0.812 + 0.583i)2-s + (−0.964 + 0.265i)3-s + (0.320 − 0.947i)4-s − 0.447i·5-s + (0.628 − 0.777i)6-s + 0.0123i·7-s + (0.292 + 0.956i)8-s + (0.858 − 0.512i)9-s + (0.260 + 0.363i)10-s + 1.56·11-s + (−0.0568 + 0.998i)12-s + 0.463·13-s + (−0.00720 − 0.0100i)14-s + (0.118 + 0.431i)15-s + (−0.795 − 0.606i)16-s − 0.825i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0568i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0568i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.998 + 0.0568i$
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: \((2,\ 60,\ (\ :3/2),\ 0.998 + 0.0568i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.753479 - 0.0214504i\)
\(L(\frac12)\) \(\approx\) \(0.753479 - 0.0214504i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.29 - 1.64i)T \)
3 \( 1 + (5.00 - 1.38i)T \)
5 \( 1 + 5iT \)
good7 \( 1 - 0.228iT - 343T^{2} \)
11 \( 1 - 57.2T + 1.33e3T^{2} \)
13 \( 1 - 21.7T + 2.19e3T^{2} \)
17 \( 1 + 57.8iT - 4.91e3T^{2} \)
19 \( 1 + 77.1iT - 6.85e3T^{2} \)
23 \( 1 - 88.7T + 1.21e4T^{2} \)
29 \( 1 + 259. iT - 2.43e4T^{2} \)
31 \( 1 - 130. iT - 2.97e4T^{2} \)
37 \( 1 - 117.T + 5.06e4T^{2} \)
41 \( 1 - 476. iT - 6.89e4T^{2} \)
43 \( 1 + 375. iT - 7.95e4T^{2} \)
47 \( 1 + 28.9T + 1.03e5T^{2} \)
53 \( 1 - 302. iT - 1.48e5T^{2} \)
59 \( 1 + 353.T + 2.05e5T^{2} \)
61 \( 1 - 593.T + 2.26e5T^{2} \)
67 \( 1 + 687. iT - 3.00e5T^{2} \)
71 \( 1 + 522.T + 3.57e5T^{2} \)
73 \( 1 + 1.16e3T + 3.89e5T^{2} \)
79 \( 1 - 884. iT - 4.93e5T^{2} \)
83 \( 1 - 390.T + 5.71e5T^{2} \)
89 \( 1 + 269. iT - 7.04e5T^{2} \)
97 \( 1 - 318.T + 9.12e5T^{2} \)
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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

−14.99823993947520728928128996828, −13.58324584934167499229602807387, −11.89495357417039009280154269443, −11.14441685162221804320400866064, −9.718857578481846676023451617741, −8.880688686447208673710455056842, −7.10205546017345098898867284092, −6.06171793130072879392021588461, −4.61431665160275764633241572793, −0.946438054529134550167896033216, 1.40707564430440712908621078292, 3.87499741225816049844746773067, 6.20395591532212041114815092934, 7.27932544029313702416112801828, 8.859848211986309998883252802119, 10.22460244709087976162347513182, 11.15239136515823096610134677891, 12.02356821587370872641683613713, 13.03254911931082977859747608374, 14.63408452377594765622833267497

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