↑

Properties

Label 2-60-60.59-c5-0-12
Degree $2$
Conductor $60$
Sign $-0.861 + 0.507i$
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  + 5.65i·2-s + (7.90 + 13.4i)3-s − 32.0·4-s + 55.9i·5-s + (−76 + 44.7i)6-s − 15.8·7-s − 181. i·8-s + (−118 + 212. i)9-s − 316.·10-s + (−252. − 429. i)12-s − 89.4i·14-s + (−751. + 441. i)15-s + 1.02e3·16-s + (−1.20e3 − 667. i)18-s − 1.78e3i·20-s + (−125 − 212. i)21-s + ⋯
L(s)  = 1  + 0.999i·2-s + (0.507 + 0.861i)3-s − 1.00·4-s + 0.999i·5-s + (−0.861 + 0.507i)6-s − 0.121·7-s − 1.00i·8-s + (−0.485 + 0.874i)9-s − 1.00·10-s + (−0.507 − 0.861i)12-s − 0.121i·14-s + (−0.861 + 0.507i)15-s + 1.00·16-s + (−0.874 − 0.485i)18-s − 1.00i·20-s + (−0.0618 − 0.105i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.861 + 0.507i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.861 + 0.507i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $-0.861 + 0.507i$
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: \((2,\ 60,\ (\ :5/2),\ -0.861 + 0.507i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.362010 - 1.32901i\)
\(L(\frac12)\) \(\approx\) \(0.362010 - 1.32901i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 5.65iT \)
3 \( 1 + (-7.90 - 13.4i)T \)
5 \( 1 - 55.9iT \)
good7 \( 1 + 15.8T + 1.68e4T^{2} \)
11 \( 1 + 1.61e5T^{2} \)
13 \( 1 - 3.71e5T^{2} \)
17 \( 1 + 1.41e6T^{2} \)
19 \( 1 - 2.47e6T^{2} \)
23 \( 1 + 3.42e3iT - 6.43e6T^{2} \)
29 \( 1 - 8.89e3iT - 2.05e7T^{2} \)
31 \( 1 - 2.86e7T^{2} \)
37 \( 1 - 6.93e7T^{2} \)
41 \( 1 - 2.10e4iT - 1.15e8T^{2} \)
43 \( 1 - 2.27e4T + 1.47e8T^{2} \)
47 \( 1 - 2.35e4iT - 2.29e8T^{2} \)
53 \( 1 + 4.18e8T^{2} \)
59 \( 1 + 7.14e8T^{2} \)
61 \( 1 - 2.54e4T + 8.44e8T^{2} \)
67 \( 1 + 1.88e4T + 1.35e9T^{2} \)
71 \( 1 + 1.80e9T^{2} \)
73 \( 1 - 2.07e9T^{2} \)
79 \( 1 - 3.07e9T^{2} \)
83 \( 1 + 1.15e5iT - 3.93e9T^{2} \)
89 \( 1 + 7.06e3iT - 5.58e9T^{2} \)
97 \( 1 - 8.58e9T^{2} \)
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

−14.57730299517027003062563615477, −14.33195915553551171410104138586, −12.88575274818001941310440145192, −10.94534066033240885499994349528, −9.939747874704541187025598821733, −8.790305351938271066563848825351, −7.53943719623145828440282174550, −6.19878858345706403182509383702, −4.60408923003764500230804365675, −3.12473285989238280777768754414, 0.62953581802254572600194155100, 2.08307634355345299263036373328, 3.86543007993307200442522082846, 5.63387835510114757447301770766, 7.71883126149496940469714314246, 8.813649404891473452226448063547, 9.747499428426657595407988920557, 11.51161670694525460683901047137, 12.35577704466322159564954005970, 13.27718540470370842283780026347

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