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Properties

Label 2-60-12.11-c5-0-38
Degree $2$
Conductor $60$
Sign $-0.853 - 0.520i$
Analytic cond. $9.62302$
Root an. cond. $3.10210$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.00749 − 5.65i)2-s + (8.15 − 13.2i)3-s + (−31.9 + 0.0848i)4-s − 25i·5-s + (−75.2 − 46.0i)6-s − 55.6i·7-s + (0.719 + 181. i)8-s + (−109. − 216. i)9-s + (−141. + 0.187i)10-s − 664.·11-s + (−259. + 425. i)12-s + 725.·13-s + (−314. + 0.416i)14-s + (−332. − 203. i)15-s + (1.02e3 − 5.42i)16-s + 42.4i·17-s + ⋯
L(s)  = 1  + (−0.00132 − 0.999i)2-s + (0.523 − 0.852i)3-s + (−0.999 + 0.00265i)4-s − 0.447i·5-s + (−0.852 − 0.522i)6-s − 0.428i·7-s + (0.00397 + 0.999i)8-s + (−0.452 − 0.891i)9-s + (−0.447 + 0.000592i)10-s − 1.65·11-s + (−0.520 + 0.853i)12-s + 1.19·13-s + (−0.428 + 0.000568i)14-s + (−0.381 − 0.233i)15-s + (0.999 − 0.00530i)16-s + 0.0356i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.853 - 0.520i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.853 - 0.520i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $-0.853 - 0.520i$
Analytic conductor: \(9.62302\)
Root analytic conductor: \(3.10210\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :5/2),\ -0.853 - 0.520i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.342295 + 1.21795i\)
\(L(\frac12)\) \(\approx\) \(0.342295 + 1.21795i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.00749 + 5.65i)T \)
3 \( 1 + (-8.15 + 13.2i)T \)
5 \( 1 + 25iT \)
good7 \( 1 + 55.6iT - 1.68e4T^{2} \)
11 \( 1 + 664.T + 1.61e5T^{2} \)
13 \( 1 - 725.T + 3.71e5T^{2} \)
17 \( 1 - 42.4iT - 1.41e6T^{2} \)
19 \( 1 - 829. iT - 2.47e6T^{2} \)
23 \( 1 + 1.46e3T + 6.43e6T^{2} \)
29 \( 1 - 365. iT - 2.05e7T^{2} \)
31 \( 1 + 7.94e3iT - 2.86e7T^{2} \)
37 \( 1 - 7.60e3T + 6.93e7T^{2} \)
41 \( 1 + 1.16e4iT - 1.15e8T^{2} \)
43 \( 1 + 2.13e4iT - 1.47e8T^{2} \)
47 \( 1 + 1.09e4T + 2.29e8T^{2} \)
53 \( 1 + 1.76e4iT - 4.18e8T^{2} \)
59 \( 1 - 2.60e4T + 7.14e8T^{2} \)
61 \( 1 + 3.94e4T + 8.44e8T^{2} \)
67 \( 1 + 2.16e3iT - 1.35e9T^{2} \)
71 \( 1 - 1.61e4T + 1.80e9T^{2} \)
73 \( 1 - 6.86e4T + 2.07e9T^{2} \)
79 \( 1 - 1.04e5iT - 3.07e9T^{2} \)
83 \( 1 + 9.25e4T + 3.93e9T^{2} \)
89 \( 1 + 1.63e3iT - 5.58e9T^{2} \)
97 \( 1 - 4.72e4T + 8.58e9T^{2} \)
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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

−13.28857866659739422082105027045, −12.51629703569470162566396891976, −11.22913822911558624316941734879, −10.02658584271657315004537051873, −8.602647629150379519981884660182, −7.76772885833646543516921307717, −5.67536017057587325248715961904, −3.77919637503985628518841335974, −2.17058521357389464875055011469, −0.56608015021981408962107939073, 3.07220254704330344039011463549, 4.74453878364847003905843391552, 6.00717983180630479709639398198, 7.76544526395299847932153469569, 8.660437583634367425310090517668, 9.925387751079857767463372169532, 10.97500205578612697976257548122, 13.01803945161833208802493288867, 13.89057006251674622388026769933, 15.00322811720547716734369143027

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