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Properties

Label 2-60-4.3-c6-0-2
Degree $2$
Conductor $60$
Sign $0.613 - 0.789i$
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  + (2.59 − 7.56i)2-s − 15.5i·3-s + (−50.5 − 39.2i)4-s − 55.9·5-s + (−117. − 40.4i)6-s + 671. i·7-s + (−428. + 280. i)8-s − 243·9-s + (−144. + 423. i)10-s − 199. i·11-s + (−611. + 787. i)12-s + 2.04e3·13-s + (5.07e3 + 1.74e3i)14-s + 871. i·15-s + (1.01e3 + 3.96e3i)16-s − 8.21e3·17-s + ⋯
L(s)  = 1  + (0.324 − 0.945i)2-s − 0.577i·3-s + (−0.789 − 0.613i)4-s − 0.447·5-s + (−0.546 − 0.187i)6-s + 1.95i·7-s + (−0.836 + 0.548i)8-s − 0.333·9-s + (−0.144 + 0.423i)10-s − 0.149i·11-s + (−0.354 + 0.455i)12-s + 0.929·13-s + (1.85 + 0.634i)14-s + 0.258i·15-s + (0.247 + 0.968i)16-s − 1.67·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.613 - 0.789i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.613 - 0.789i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.613 - 0.789i$
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: \((2,\ 60,\ (\ :3),\ 0.613 - 0.789i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.692038 + 0.338779i\)
\(L(\frac12)\) \(\approx\) \(0.692038 + 0.338779i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.59 + 7.56i)T \)
3 \( 1 + 15.5iT \)
5 \( 1 + 55.9T \)
good7 \( 1 - 671. iT - 1.17e5T^{2} \)
11 \( 1 + 199. iT - 1.77e6T^{2} \)
13 \( 1 - 2.04e3T + 4.82e6T^{2} \)
17 \( 1 + 8.21e3T + 2.41e7T^{2} \)
19 \( 1 + 5.14e3iT - 4.70e7T^{2} \)
23 \( 1 - 1.46e4iT - 1.48e8T^{2} \)
29 \( 1 + 1.25e4T + 5.94e8T^{2} \)
31 \( 1 - 5.44e4iT - 8.87e8T^{2} \)
37 \( 1 + 11.6T + 2.56e9T^{2} \)
41 \( 1 + 6.78e4T + 4.75e9T^{2} \)
43 \( 1 - 1.05e5iT - 6.32e9T^{2} \)
47 \( 1 + 8.96e4iT - 1.07e10T^{2} \)
53 \( 1 + 1.36e3T + 2.21e10T^{2} \)
59 \( 1 - 5.60e3iT - 4.21e10T^{2} \)
61 \( 1 + 1.32e5T + 5.15e10T^{2} \)
67 \( 1 + 3.25e5iT - 9.04e10T^{2} \)
71 \( 1 + 3.03e5iT - 1.28e11T^{2} \)
73 \( 1 + 2.47e5T + 1.51e11T^{2} \)
79 \( 1 - 3.89e5iT - 2.43e11T^{2} \)
83 \( 1 - 2.15e5iT - 3.26e11T^{2} \)
89 \( 1 - 4.62e4T + 4.96e11T^{2} \)
97 \( 1 - 8.00e5T + 8.32e11T^{2} \)
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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

−13.62723356125158881743752545346, −12.74522647542493145969450932739, −11.73897760787068969845605301073, −11.09793606788059864314596404715, −9.132175041051086414067338960092, −8.534204295508521910653208144905, −6.34649563706348074811178066735, −5.08031935222318889165560351904, −3.12893051284227983984114338065, −1.80597965184834123350176174419, 0.28526202410606215602659275119, 3.82297276831820406617990289795, 4.44472742270473626821805483510, 6.37694994025025738577839294214, 7.50632930073835848599110188679, 8.685251421407802100738015967769, 10.19616161623005687452322071361, 11.26521212060027680143595648595, 13.05070609731510891999454656072, 13.79699808235389448604713423409

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