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Properties

Label 2-60-4.3-c6-0-0
Degree $2$
Conductor $60$
Sign $-0.711 - 0.702i$
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  + (−3.08 − 7.38i)2-s + 15.5i·3-s + (−44.9 + 45.5i)4-s + 55.9·5-s + (115. − 48.0i)6-s − 200. i·7-s + (474. + 191. i)8-s − 243·9-s + (−172. − 412. i)10-s + 76.4i·11-s + (−709. − 701. i)12-s − 2.21e3·13-s + (−1.47e3 + 616. i)14-s + 871. i·15-s + (−51.2 − 4.09e3i)16-s − 7.76e3·17-s + ⋯
L(s)  = 1  + (−0.385 − 0.922i)2-s + 0.577i·3-s + (−0.702 + 0.711i)4-s + 0.447·5-s + (0.532 − 0.222i)6-s − 0.583i·7-s + (0.927 + 0.373i)8-s − 0.333·9-s + (−0.172 − 0.412i)10-s + 0.0574i·11-s + (−0.410 − 0.405i)12-s − 1.00·13-s + (−0.538 + 0.224i)14-s + 0.258i·15-s + (−0.0125 − 0.999i)16-s − 1.58·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.711 - 0.702i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.711 - 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $-0.711 - 0.702i$
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: \((2,\ 60,\ (\ :3),\ -0.711 - 0.702i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.0577084 + 0.140562i\)
\(L(\frac12)\) \(\approx\) \(0.0577084 + 0.140562i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (3.08 + 7.38i)T \)
3 \( 1 - 15.5iT \)
5 \( 1 - 55.9T \)
good7 \( 1 + 200. iT - 1.17e5T^{2} \)
11 \( 1 - 76.4iT - 1.77e6T^{2} \)
13 \( 1 + 2.21e3T + 4.82e6T^{2} \)
17 \( 1 + 7.76e3T + 2.41e7T^{2} \)
19 \( 1 - 3.20e3iT - 4.70e7T^{2} \)
23 \( 1 - 1.24e4iT - 1.48e8T^{2} \)
29 \( 1 + 3.66e4T + 5.94e8T^{2} \)
31 \( 1 - 1.37e4iT - 8.87e8T^{2} \)
37 \( 1 + 4.40e4T + 2.56e9T^{2} \)
41 \( 1 + 7.55e4T + 4.75e9T^{2} \)
43 \( 1 + 2.22e4iT - 6.32e9T^{2} \)
47 \( 1 + 1.76e5iT - 1.07e10T^{2} \)
53 \( 1 + 7.50e4T + 2.21e10T^{2} \)
59 \( 1 - 1.76e5iT - 4.21e10T^{2} \)
61 \( 1 + 3.26e4T + 5.15e10T^{2} \)
67 \( 1 + 3.09e5iT - 9.04e10T^{2} \)
71 \( 1 - 3.77e5iT - 1.28e11T^{2} \)
73 \( 1 - 4.23e5T + 1.51e11T^{2} \)
79 \( 1 - 6.43e5iT - 2.43e11T^{2} \)
83 \( 1 + 8.24e5iT - 3.26e11T^{2} \)
89 \( 1 + 5.39e5T + 4.96e11T^{2} \)
97 \( 1 - 1.03e6T + 8.32e11T^{2} \)
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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

−13.99359044715980599530833715996, −13.10374733722919440086405782575, −11.77310790744001887049394882874, −10.68148199949404321452882800461, −9.808092482335865337932739161519, −8.805376920641855206167422957976, −7.23300403625698617343885844360, −5.07696787236443694232204480243, −3.71762531176009290729383391878, −2.01001734446990502371842469734, 0.06710743361347484847893246473, 2.12481469327913397539378418829, 4.84940487215523981882833026041, 6.18113785622561913874921895439, 7.21855162636375975663655462418, 8.597480582554934518303586782398, 9.522689162633124479904382542845, 11.00337157142225849848484178569, 12.60419712806468916506737933290, 13.56515856490184136853151225989

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