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Properties

Label 2-60-4.3-c6-0-20
Degree $2$
Conductor $60$
Sign $0.0317 + 0.999i$
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  + (7.99 − 0.126i)2-s − 15.5i·3-s + (63.9 − 2.02i)4-s − 55.9·5-s + (−1.97 − 124. i)6-s − 335. i·7-s + (511. − 24.3i)8-s − 243·9-s + (−447. + 7.09i)10-s − 1.64e3i·11-s + (−31.6 − 997. i)12-s + 1.20e3·13-s + (−42.5 − 2.68e3i)14-s + 871. i·15-s + (4.08e3 − 259. i)16-s − 3.94e3·17-s + ⋯
L(s)  = 1  + (0.999 − 0.0158i)2-s − 0.577i·3-s + (0.999 − 0.0317i)4-s − 0.447·5-s + (−0.00915 − 0.577i)6-s − 0.977i·7-s + (0.998 − 0.0475i)8-s − 0.333·9-s + (−0.447 + 0.00709i)10-s − 1.23i·11-s + (−0.0183 − 0.577i)12-s + 0.546·13-s + (−0.0154 − 0.977i)14-s + 0.258i·15-s + (0.997 − 0.0633i)16-s − 0.802·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0317 + 0.999i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.0317 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.0317 + 0.999i$
Analytic conductor: \(13.8032\)
Root analytic conductor: \(3.71527\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :3),\ 0.0317 + 0.999i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(2.18210 - 2.11396i\)
\(L(\frac12)\) \(\approx\) \(2.18210 - 2.11396i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-7.99 + 0.126i)T \)
3 \( 1 + 15.5iT \)
5 \( 1 + 55.9T \)
good7 \( 1 + 335. iT - 1.17e5T^{2} \)
11 \( 1 + 1.64e3iT - 1.77e6T^{2} \)
13 \( 1 - 1.20e3T + 4.82e6T^{2} \)
17 \( 1 + 3.94e3T + 2.41e7T^{2} \)
19 \( 1 + 3.73e3iT - 4.70e7T^{2} \)
23 \( 1 + 4.91e3iT - 1.48e8T^{2} \)
29 \( 1 - 2.03e4T + 5.94e8T^{2} \)
31 \( 1 - 3.84e4iT - 8.87e8T^{2} \)
37 \( 1 - 1.76e4T + 2.56e9T^{2} \)
41 \( 1 - 5.41e4T + 4.75e9T^{2} \)
43 \( 1 - 3.92e3iT - 6.32e9T^{2} \)
47 \( 1 - 1.45e5iT - 1.07e10T^{2} \)
53 \( 1 + 2.23e5T + 2.21e10T^{2} \)
59 \( 1 - 3.49e5iT - 4.21e10T^{2} \)
61 \( 1 - 1.44e5T + 5.15e10T^{2} \)
67 \( 1 - 5.24e5iT - 9.04e10T^{2} \)
71 \( 1 + 2.02e5iT - 1.28e11T^{2} \)
73 \( 1 - 6.74e5T + 1.51e11T^{2} \)
79 \( 1 + 3.11e5iT - 2.43e11T^{2} \)
83 \( 1 + 9.75e5iT - 3.26e11T^{2} \)
89 \( 1 + 1.14e6T + 4.96e11T^{2} \)
97 \( 1 - 5.73e5T + 8.32e11T^{2} \)
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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

−13.61733932934878696474574679492, −12.68148886879937929195740257880, −11.36122370736708078475470316146, −10.70520126647119557493703219470, −8.473618437404801165560298331077, −7.19127745735602909237306420349, −6.14699197501182577318223939856, −4.42493001168466585704836851536, −3.04142554305893918843183635446, −0.958877769104565423851515474109, 2.26160125933684138518343104100, 3.90126134201507879841274643442, 5.09402583588512633037767855107, 6.46148897093615043959983619157, 8.033798969023613558435375027476, 9.597140269525392595453907550713, 11.01105647616331757758704887209, 11.97748296904117729876409344455, 12.89895379209385315504808854464, 14.29242109319576329412779374871

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