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Properties

Label 2-60-4.3-c4-0-8
Degree $2$
Conductor $60$
Sign $0.460 - 0.887i$
Analytic cond. $6.20219$
Root an. cond. $2.49042$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.88 + 0.947i)2-s + 5.19i·3-s + (14.2 + 7.36i)4-s + 11.1·5-s + (−4.92 + 20.1i)6-s + 12.7i·7-s + (48.2 + 42.0i)8-s − 27·9-s + (43.4 + 10.5i)10-s + 45.0i·11-s + (−38.2 + 73.8i)12-s + 1.08·13-s + (−12.1 + 49.6i)14-s + 58.0i·15-s + (147. + 209. i)16-s + 40.6·17-s + ⋯
L(s)  = 1  + (0.971 + 0.236i)2-s + 0.577i·3-s + (0.887 + 0.460i)4-s + 0.447·5-s + (−0.136 + 0.560i)6-s + 0.260i·7-s + (0.753 + 0.657i)8-s − 0.333·9-s + (0.434 + 0.105i)10-s + 0.372i·11-s + (−0.265 + 0.512i)12-s + 0.00642·13-s + (−0.0617 + 0.253i)14-s + 0.258i·15-s + (0.576 + 0.817i)16-s + 0.140·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.460 - 0.887i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.460 - 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.460 - 0.887i$
Analytic conductor: \(6.20219\)
Root analytic conductor: \(2.49042\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :2),\ 0.460 - 0.887i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.49776 + 1.51842i\)
\(L(\frac12)\) \(\approx\) \(2.49776 + 1.51842i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-3.88 - 0.947i)T \)
3 \( 1 - 5.19iT \)
5 \( 1 - 11.1T \)
good7 \( 1 - 12.7iT - 2.40e3T^{2} \)
11 \( 1 - 45.0iT - 1.46e4T^{2} \)
13 \( 1 - 1.08T + 2.85e4T^{2} \)
17 \( 1 - 40.6T + 8.35e4T^{2} \)
19 \( 1 + 290. iT - 1.30e5T^{2} \)
23 \( 1 + 949. iT - 2.79e5T^{2} \)
29 \( 1 + 402.T + 7.07e5T^{2} \)
31 \( 1 + 762. iT - 9.23e5T^{2} \)
37 \( 1 - 1.32e3T + 1.87e6T^{2} \)
41 \( 1 + 3.20e3T + 2.82e6T^{2} \)
43 \( 1 + 2.38e3iT - 3.41e6T^{2} \)
47 \( 1 - 730. iT - 4.87e6T^{2} \)
53 \( 1 + 3.35e3T + 7.89e6T^{2} \)
59 \( 1 - 5.21e3iT - 1.21e7T^{2} \)
61 \( 1 - 4.69e3T + 1.38e7T^{2} \)
67 \( 1 - 3.35e3iT - 2.01e7T^{2} \)
71 \( 1 - 8.48e3iT - 2.54e7T^{2} \)
73 \( 1 + 174.T + 2.83e7T^{2} \)
79 \( 1 - 1.01e4iT - 3.89e7T^{2} \)
83 \( 1 + 8.83e3iT - 4.74e7T^{2} \)
89 \( 1 + 1.93e3T + 6.27e7T^{2} \)
97 \( 1 + 1.42e4T + 8.85e7T^{2} \)
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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

−14.62570320775701369162768083690, −13.51020494543651509566713643194, −12.44982970861798494275631961535, −11.25642552242024170940651068518, −10.07812608993209672425029356471, −8.557832123035008400711666889020, −6.88484091130025768846735956681, −5.55724386042459026711377538161, −4.31078391859186359851955494739, −2.54591224914774597710772893055, 1.58161136103168396973474488632, 3.40941492923467590884596407696, 5.30870883043610224937328708074, 6.47303764519994594022603624640, 7.80350187639481870410347687230, 9.710935658991049378763283024871, 11.01862295804187140873592573151, 12.06488445530554550947789959141, 13.19210517393561134321038991737, 13.87936487787347517065956405160

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