↑

Properties

Label 2-60-60.59-c5-0-49
Degree $2$
Conductor $60$
Sign $0.441 + 0.897i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (5.20 − 2.22i)2-s + (15.5 + 0.545i)3-s + (22.1 − 23.1i)4-s + (17.3 − 53.1i)5-s + (82.2 − 31.7i)6-s − 106.·7-s + (63.7 − 169. i)8-s + (242. + 16.9i)9-s + (−27.9 − 314. i)10-s − 278.·11-s + (357. − 348. i)12-s + 699. i·13-s + (−552. + 235. i)14-s + (298. − 818. i)15-s + (−44.7 − 1.02e3i)16-s + 1.73e3·17-s + ⋯
L(s)  = 1  + (0.919 − 0.392i)2-s + (0.999 + 0.0349i)3-s + (0.691 − 0.722i)4-s + (0.309 − 0.950i)5-s + (0.932 − 0.360i)6-s − 0.819·7-s + (0.352 − 0.935i)8-s + (0.997 + 0.0699i)9-s + (−0.0885 − 0.996i)10-s − 0.692·11-s + (0.716 − 0.697i)12-s + 1.14i·13-s + (−0.753 + 0.321i)14-s + (0.342 − 0.939i)15-s + (−0.0436 − 0.999i)16-s + 1.45·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.441 + 0.897i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.441 + 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.441 + 0.897i$
Analytic conductor: \(9.62302\)
Root analytic conductor: \(3.10210\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :5/2),\ 0.441 + 0.897i)\)

Particular Values

\(L(3)\) \(\approx\) \(3.28008 - 2.04172i\)
\(L(\frac12)\) \(\approx\) \(3.28008 - 2.04172i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-5.20 + 2.22i)T \)
3 \( 1 + (-15.5 - 0.545i)T \)
5 \( 1 + (-17.3 + 53.1i)T \)
good7 \( 1 + 106.T + 1.68e4T^{2} \)
11 \( 1 + 278.T + 1.61e5T^{2} \)
13 \( 1 - 699. iT - 3.71e5T^{2} \)
17 \( 1 - 1.73e3T + 1.41e6T^{2} \)
19 \( 1 - 2.22e3iT - 2.47e6T^{2} \)
23 \( 1 + 1.75e3iT - 6.43e6T^{2} \)
29 \( 1 + 2.52e3iT - 2.05e7T^{2} \)
31 \( 1 - 5.69e3iT - 2.86e7T^{2} \)
37 \( 1 - 977. iT - 6.93e7T^{2} \)
41 \( 1 - 1.74e4iT - 1.15e8T^{2} \)
43 \( 1 - 5.03e3T + 1.47e8T^{2} \)
47 \( 1 - 4.80e3iT - 2.29e8T^{2} \)
53 \( 1 + 2.72e4T + 4.18e8T^{2} \)
59 \( 1 + 2.28e4T + 7.14e8T^{2} \)
61 \( 1 + 1.22e4T + 8.44e8T^{2} \)
67 \( 1 - 4.79e4T + 1.35e9T^{2} \)
71 \( 1 + 3.01e4T + 1.80e9T^{2} \)
73 \( 1 + 6.31e4iT - 2.07e9T^{2} \)
79 \( 1 + 8.40e4iT - 3.07e9T^{2} \)
83 \( 1 + 5.67e4iT - 3.93e9T^{2} \)
89 \( 1 - 1.19e4iT - 5.58e9T^{2} \)
97 \( 1 - 2.38e4iT - 8.58e9T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.84332769477723625275186472084, −12.82329998234126468602648214479, −12.18146538830502941078216175988, −10.21119511341320252706022313406, −9.450568203437839428048787469344, −7.891091148470620615031981451221, −6.19772416279380065634068949498, −4.62153127315919836034309348715, −3.22896683368615618469067678832, −1.58589966002843672041877227215, 2.66176890593874030505836289200, 3.45778605685781574936661254555, 5.54757491026340430758845756991, 7.01373414778064675164542792514, 7.902446378415267187306005141068, 9.676207980055080649553193008655, 10.82897989352928057378333895332, 12.57995823383468867640703927322, 13.36161086124606163070483805846, 14.24072710094964228126463476109

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