Properties

Label 2-60-15.14-c6-0-9
Degree $2$
Conductor $60$
Sign $-0.286 + 0.957i$
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  + (22.8 − 14.3i)3-s + (−93.9 + 82.4i)5-s − 279. i·7-s + (318. − 655. i)9-s − 772. i·11-s − 2.87e3i·13-s + (−968. + 3.23e3i)15-s + 2.79e3·17-s − 1.30e4·19-s + (−4.00e3 − 6.40e3i)21-s + 1.05e4·23-s + (2.02e3 − 1.54e4i)25-s + (−2.10e3 − 1.95e4i)27-s − 3.72e4i·29-s − 8.71e3·31-s + ⋯
L(s)  = 1  + (0.847 − 0.530i)3-s + (−0.751 + 0.659i)5-s − 0.815i·7-s + (0.436 − 0.899i)9-s − 0.580i·11-s − 1.30i·13-s + (−0.286 + 0.957i)15-s + 0.568·17-s − 1.90·19-s + (−0.432 − 0.691i)21-s + 0.867·23-s + (0.129 − 0.991i)25-s + (−0.107 − 0.994i)27-s − 1.52i·29-s − 0.292·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.286 + 0.957i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.286 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $-0.286 + 0.957i$
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: \((2,\ 60,\ (\ :3),\ -0.286 + 0.957i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.01896 - 1.36882i\)
\(L(\frac12)\) \(\approx\) \(1.01896 - 1.36882i\)
\(L(4)\) 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 \( 1 + (-22.8 + 14.3i)T \)
5 \( 1 + (93.9 - 82.4i)T \)
good7 \( 1 + 279. iT - 1.17e5T^{2} \)
11 \( 1 + 772. iT - 1.77e6T^{2} \)
13 \( 1 + 2.87e3iT - 4.82e6T^{2} \)
17 \( 1 - 2.79e3T + 2.41e7T^{2} \)
19 \( 1 + 1.30e4T + 4.70e7T^{2} \)
23 \( 1 - 1.05e4T + 1.48e8T^{2} \)
29 \( 1 + 3.72e4iT - 5.94e8T^{2} \)
31 \( 1 + 8.71e3T + 8.87e8T^{2} \)
37 \( 1 + 3.55e3iT - 2.56e9T^{2} \)
41 \( 1 - 1.08e5iT - 4.75e9T^{2} \)
43 \( 1 - 6.66e4iT - 6.32e9T^{2} \)
47 \( 1 + 4.02e3T + 1.07e10T^{2} \)
53 \( 1 - 7.95e4T + 2.21e10T^{2} \)
59 \( 1 - 1.51e5iT - 4.21e10T^{2} \)
61 \( 1 - 2.07e5T + 5.15e10T^{2} \)
67 \( 1 + 3.63e5iT - 9.04e10T^{2} \)
71 \( 1 - 3.79e4iT - 1.28e11T^{2} \)
73 \( 1 - 4.61e5iT - 1.51e11T^{2} \)
79 \( 1 + 2.27e5T + 2.43e11T^{2} \)
83 \( 1 - 9.98e5T + 3.26e11T^{2} \)
89 \( 1 - 1.03e6iT - 4.96e11T^{2} \)
97 \( 1 + 7.34e5iT - 8.32e11T^{2} \)
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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

−13.50691756940958679445086374074, −12.63202653504152006543610423348, −11.15555663388928933871868518108, −10.11768960767278263493138331017, −8.378631767951237154462022763704, −7.63396151974655965279100902987, −6.39349252578967442748428417014, −4.00998985948807334582395709799, −2.81964337680800346491339616358, −0.64135552506006148321730636308, 2.02779074027549781610676541851, 3.85177537198930116458037089091, 5.00624005020448084973055478232, 7.13006903136118081080862078988, 8.628927243895514125748142282758, 9.101753224012212991096065115736, 10.67307151470598575850732428227, 12.07089868389410379341117229988, 12.97281484749765220435857215306, 14.47859058851753727856079882174

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