↑

Properties

Label 2-60-60.59-c5-0-32
Degree $2$
Conductor $60$
Sign $0.915 + 0.401i$
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 + (−79.8 + 37.4i)6-s + 106.·7-s + (63.7 − 169. i)8-s + (242. − 16.9i)9-s + (208. + 238. i)10-s + 278.·11-s + (−332. + 372. 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 + (−0.905 + 0.424i)6-s + 0.819·7-s + (0.352 − 0.935i)8-s + (0.997 − 0.0699i)9-s + (0.658 + 0.752i)10-s + 0.692·11-s + (−0.665 + 0.746i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.915 + 0.401i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.915 + 0.401i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.915 + 0.401i$
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: \((2,\ 60,\ (\ :5/2),\ 0.915 + 0.401i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.55083 - 0.535083i\)
\(L(\frac12)\) \(\approx\) \(2.55083 - 0.535083i\)
\(L(\frac{7}{2})\) 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

−14.22249622302719572783867974147, −12.61800398930617997221041001196, −11.83552292299038432425184916021, −10.68064150216146666936454578343, −10.12949647651918939297122334867, −7.52170011732312306534782714157, −6.17963731066516996431470223215, −5.23220543660925254504971976610, −3.52341093781575273791800211522, −1.43780815090803347279261177473, 1.49210490316601111431130252837, 4.31437213742781407300765032377, 5.21170016929218176150045854168, 6.42923440853490046393481794022, 7.83641914931480673560102519066, 9.495674540951202517334647452586, 11.40182904709595217685579192850, 11.84561667827951419915588330037, 13.04206732374306598534206881810, 14.03379073179607080283027813331

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