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Properties

Label 2-60-15.14-c6-0-1
Degree $2$
Conductor $60$
Sign $-0.999 - 0.0149i$
Analytic cond. $13.8032$
Root an. cond. $3.71527$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (7.66 + 25.8i)3-s + (−37.2 + 119. i)5-s + 46.7i·7-s + (−611. + 396. i)9-s + 448. i·11-s − 2.07e3i·13-s + (−3.37e3 − 50.5i)15-s − 5.98e3·17-s + 7.40e3·19-s + (−1.20e3 + 358. i)21-s − 1.71e4·23-s + (−1.28e4 − 8.89e3i)25-s + (−1.49e4 − 1.27e4i)27-s + 3.75e4i·29-s − 1.79e4·31-s + ⋯
L(s)  = 1  + (0.283 + 0.958i)3-s + (−0.298 + 0.954i)5-s + 0.136i·7-s + (−0.838 + 0.544i)9-s + 0.337i·11-s − 0.943i·13-s + (−0.999 − 0.0149i)15-s − 1.21·17-s + 1.07·19-s + (−0.130 + 0.0386i)21-s − 1.41·23-s + (−0.822 − 0.569i)25-s + (−0.759 − 0.649i)27-s + 1.53i·29-s − 0.603·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0149i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.999 - 0.0149i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $-0.999 - 0.0149i$
Analytic conductor: \(13.8032\)
Root analytic conductor: \(3.71527\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :3),\ -0.999 - 0.0149i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.00799827 + 1.06822i\)
\(L(\frac12)\) \(\approx\) \(0.00799827 + 1.06822i\)
\(L(4)\) 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 \( 1 + (-7.66 - 25.8i)T \)
5 \( 1 + (37.2 - 119. i)T \)
good7 \( 1 - 46.7iT - 1.17e5T^{2} \)
11 \( 1 - 448. iT - 1.77e6T^{2} \)
13 \( 1 + 2.07e3iT - 4.82e6T^{2} \)
17 \( 1 + 5.98e3T + 2.41e7T^{2} \)
19 \( 1 - 7.40e3T + 4.70e7T^{2} \)
23 \( 1 + 1.71e4T + 1.48e8T^{2} \)
29 \( 1 - 3.75e4iT - 5.94e8T^{2} \)
31 \( 1 + 1.79e4T + 8.87e8T^{2} \)
37 \( 1 - 3.62e4iT - 2.56e9T^{2} \)
41 \( 1 - 1.28e4iT - 4.75e9T^{2} \)
43 \( 1 + 2.50e4iT - 6.32e9T^{2} \)
47 \( 1 - 5.18e4T + 1.07e10T^{2} \)
53 \( 1 - 2.70e5T + 2.21e10T^{2} \)
59 \( 1 - 2.18e5iT - 4.21e10T^{2} \)
61 \( 1 - 3.83e4T + 5.15e10T^{2} \)
67 \( 1 - 2.59e5iT - 9.04e10T^{2} \)
71 \( 1 - 4.52e5iT - 1.28e11T^{2} \)
73 \( 1 + 7.84e4iT - 1.51e11T^{2} \)
79 \( 1 + 4.10e5T + 2.43e11T^{2} \)
83 \( 1 + 6.30e5T + 3.26e11T^{2} \)
89 \( 1 - 1.45e5iT - 4.96e11T^{2} \)
97 \( 1 - 1.76e6iT - 8.32e11T^{2} \)
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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

−14.61682834479970137571436827837, −13.56368511052505057051623424793, −11.90039561658199465198573245539, −10.77735819528588490158590111097, −9.968795967667939935566300821634, −8.596922243214156232206296358727, −7.24846506470169002873502422418, −5.56134594513863441653586083439, −3.94544963115327861372820595598, −2.64927072104145243425493137691, 0.40983004168409188473700010386, 1.99330291030442800136958002334, 4.07781222963225949092421570834, 5.86074041549824513210582859724, 7.30459249351420387907840189182, 8.439245356617273974504397279804, 9.423851263053981566914977825963, 11.41860593099302284715245466415, 12.17476604245462695254654437245, 13.38163872687343535327097275413

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