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Properties

Label 2-60-3.2-c6-0-5
Degree $2$
Conductor $60$
Sign $-0.570 + 0.821i$
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  + (−22.1 − 15.3i)3-s + 55.9i·5-s + 437.·7-s + (254. + 683. i)9-s − 1.89e3i·11-s − 2.67e3·13-s + (860. − 1.23e3i)15-s − 4.52e3i·17-s − 9.50e3·19-s + (−9.69e3 − 6.73e3i)21-s − 549. i·23-s − 3.12e3·25-s + (4.86e3 − 1.90e4i)27-s − 4.33e4i·29-s + 2.95e4·31-s + ⋯
L(s)  = 1  + (−0.821 − 0.570i)3-s + 0.447i·5-s + 1.27·7-s + (0.349 + 0.936i)9-s − 1.42i·11-s − 1.21·13-s + (0.255 − 0.367i)15-s − 0.920i·17-s − 1.38·19-s + (−1.04 − 0.726i)21-s − 0.0451i·23-s − 0.199·25-s + (0.247 − 0.968i)27-s − 1.77i·29-s + 0.993·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.570 + 0.821i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.570 + 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $-0.570 + 0.821i$
Analytic conductor: \(13.8032\)
Root analytic conductor: \(3.71527\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :3),\ -0.570 + 0.821i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.433337 - 0.828447i\)
\(L(\frac12)\) \(\approx\) \(0.433337 - 0.828447i\)
\(L(4)\) 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 \( 1 + (22.1 + 15.3i)T \)
5 \( 1 - 55.9iT \)
good7 \( 1 - 437.T + 1.17e5T^{2} \)
11 \( 1 + 1.89e3iT - 1.77e6T^{2} \)
13 \( 1 + 2.67e3T + 4.82e6T^{2} \)
17 \( 1 + 4.52e3iT - 2.41e7T^{2} \)
19 \( 1 + 9.50e3T + 4.70e7T^{2} \)
23 \( 1 + 549. iT - 1.48e8T^{2} \)
29 \( 1 + 4.33e4iT - 5.94e8T^{2} \)
31 \( 1 - 2.95e4T + 8.87e8T^{2} \)
37 \( 1 + 4.57e4T + 2.56e9T^{2} \)
41 \( 1 + 1.14e5iT - 4.75e9T^{2} \)
43 \( 1 + 2.49e4T + 6.32e9T^{2} \)
47 \( 1 - 1.82e4iT - 1.07e10T^{2} \)
53 \( 1 - 1.74e5iT - 2.21e10T^{2} \)
59 \( 1 + 6.78e4iT - 4.21e10T^{2} \)
61 \( 1 - 1.75e5T + 5.15e10T^{2} \)
67 \( 1 - 1.60e4T + 9.04e10T^{2} \)
71 \( 1 - 2.22e5iT - 1.28e11T^{2} \)
73 \( 1 + 6.20e5T + 1.51e11T^{2} \)
79 \( 1 - 5.25e5T + 2.43e11T^{2} \)
83 \( 1 + 7.90e5iT - 3.26e11T^{2} \)
89 \( 1 + 1.12e6iT - 4.96e11T^{2} \)
97 \( 1 + 6.24e5T + 8.32e11T^{2} \)
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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

−13.54669357874543182905157654725, −12.03419597061837325688130707860, −11.32228508383665672872214678995, −10.35115237668116539786257050942, −8.413983276345194522696231108137, −7.32140042304397597268494850576, −5.95114865332614676188591458183, −4.67416677816333526884305291783, −2.27241430403031515503595004021, −0.42633717453504609896932910238, 1.69700472910962308577061090316, 4.42910395939527526405803560253, 5.08801394335289694242771707232, 6.85601454902452523449652325997, 8.353948134512688050134692949215, 9.813128644148999050319537140315, 10.77595834830172876679979775755, 12.02157394598668942950851817047, 12.71693162506461125765378148855, 14.75072157735704628171702222595

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