↑

Properties

Label 2-60-60.23-c2-0-19
Degree $2$
Conductor $60$
Sign $-0.664 - 0.747i$
Analytic cond. $1.63488$
Root an. cond. $1.27862$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.837 − 1.81i)2-s + (−2.69 − 1.32i)3-s + (−2.59 + 3.04i)4-s + (−3.21 + 3.82i)5-s + (−0.144 + 5.99i)6-s + (−3.54 − 3.54i)7-s + (7.70 + 2.16i)8-s + (5.50 + 7.12i)9-s + (9.64 + 2.63i)10-s − 16.8·11-s + (11.0 − 4.76i)12-s + (−8.64 − 8.64i)13-s + (−3.46 + 9.40i)14-s + (13.7 − 6.06i)15-s + (−2.51 − 15.8i)16-s + (−9.72 − 9.72i)17-s + ⋯
L(s)  = 1  + (−0.418 − 0.908i)2-s + (−0.897 − 0.440i)3-s + (−0.649 + 0.760i)4-s + (−0.642 + 0.765i)5-s + (−0.0241 + 0.999i)6-s + (−0.506 − 0.506i)7-s + (0.962 + 0.270i)8-s + (0.611 + 0.791i)9-s + (0.964 + 0.263i)10-s − 1.53·11-s + (0.917 − 0.396i)12-s + (−0.665 − 0.665i)13-s + (−0.247 + 0.671i)14-s + (0.914 − 0.404i)15-s + (−0.157 − 0.987i)16-s + (−0.572 − 0.572i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.664 - 0.747i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.664 - 0.747i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $-0.664 - 0.747i$
Analytic conductor: \(1.63488\)
Root analytic conductor: \(1.27862\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :1),\ -0.664 - 0.747i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0239528 + 0.0533605i\)
\(L(\frac12)\) \(\approx\) \(0.0239528 + 0.0533605i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.837 + 1.81i)T \)
3 \( 1 + (2.69 + 1.32i)T \)
5 \( 1 + (3.21 - 3.82i)T \)
good7 \( 1 + (3.54 + 3.54i)T + 49iT^{2} \)
11 \( 1 + 16.8T + 121T^{2} \)
13 \( 1 + (8.64 + 8.64i)T + 169iT^{2} \)
17 \( 1 + (9.72 + 9.72i)T + 289iT^{2} \)
19 \( 1 - 4.78T + 361T^{2} \)
23 \( 1 + (-13.5 - 13.5i)T + 529iT^{2} \)
29 \( 1 - 14.8T + 841T^{2} \)
31 \( 1 - 14.0iT - 961T^{2} \)
37 \( 1 + (10.1 - 10.1i)T - 1.36e3iT^{2} \)
41 \( 1 + 6.08iT - 1.68e3T^{2} \)
43 \( 1 + (57.2 - 57.2i)T - 1.84e3iT^{2} \)
47 \( 1 + (-17.6 + 17.6i)T - 2.20e3iT^{2} \)
53 \( 1 + (16.2 - 16.2i)T - 2.80e3iT^{2} \)
59 \( 1 + 4.37iT - 3.48e3T^{2} \)
61 \( 1 - 8.52T + 3.72e3T^{2} \)
67 \( 1 + (53.9 + 53.9i)T + 4.48e3iT^{2} \)
71 \( 1 + 36.6T + 5.04e3T^{2} \)
73 \( 1 + (12.6 + 12.6i)T + 5.32e3iT^{2} \)
79 \( 1 - 88.4T + 6.24e3T^{2} \)
83 \( 1 + (63.7 + 63.7i)T + 6.88e3iT^{2} \)
89 \( 1 + 115.T + 7.92e3T^{2} \)
97 \( 1 + (85.3 - 85.3i)T - 9.40e3iT^{2} \)
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

−13.61917448041106214554447723143, −12.80232629073491961807651988358, −11.70018029504377302419875850940, −10.71497212581052471531904879239, −10.04551831097440500857501663721, −7.926285739117201535514024664287, −7.05701129158171076245551159394, −4.93474462113777815927096236843, −2.92921736848976104090225264098, −0.06749241208138889012067653804, 4.55221064453223841022198221380, 5.56415824582306887973205862493, 7.03628069254276070599085091198, 8.477768691586605376314847852987, 9.637467110675315113405936930105, 10.80132294296609271914667437346, 12.26237403429640359544693759531, 13.21941318419961090733009096579, 15.07612862079062065234004866813, 15.74224386791946577260331834763

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