↑

Properties

Label 2-60-60.59-c5-0-44
Degree $2$
Conductor $60$
Sign $0.105 + 0.994i$
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  + (3.08 − 4.73i)2-s + (14.4 − 5.82i)3-s + (−12.9 − 29.2i)4-s + (43.0 + 35.7i)5-s + (17.0 − 86.5i)6-s + 168.·7-s + (−178. − 29.2i)8-s + (175. − 168. i)9-s + (302. − 93.5i)10-s − 165.·11-s + (−357. − 348. i)12-s − 182. i·13-s + (519. − 796. i)14-s + (829. + 265. i)15-s + (−690. + 756. i)16-s − 1.80e3·17-s + ⋯
L(s)  = 1  + (0.546 − 0.837i)2-s + (0.927 − 0.373i)3-s + (−0.403 − 0.914i)4-s + (0.769 + 0.638i)5-s + (0.193 − 0.981i)6-s + 1.29·7-s + (−0.986 − 0.161i)8-s + (0.720 − 0.693i)9-s + (0.955 − 0.295i)10-s − 0.412·11-s + (−0.716 − 0.697i)12-s − 0.299i·13-s + (0.708 − 1.08i)14-s + (0.952 + 0.304i)15-s + (−0.674 + 0.738i)16-s − 1.51·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.105 + 0.994i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.105 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.105 + 0.994i$
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: \((2,\ 60,\ (\ :5/2),\ 0.105 + 0.994i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.51227 - 2.26043i\)
\(L(\frac12)\) \(\approx\) \(2.51227 - 2.26043i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-3.08 + 4.73i)T \)
3 \( 1 + (-14.4 + 5.82i)T \)
5 \( 1 + (-43.0 - 35.7i)T \)
good7 \( 1 - 168.T + 1.68e4T^{2} \)
11 \( 1 + 165.T + 1.61e5T^{2} \)
13 \( 1 + 182. iT - 3.71e5T^{2} \)
17 \( 1 + 1.80e3T + 1.41e6T^{2} \)
19 \( 1 - 1.17e3iT - 2.47e6T^{2} \)
23 \( 1 + 391. iT - 6.43e6T^{2} \)
29 \( 1 + 382. iT - 2.05e7T^{2} \)
31 \( 1 + 8.73e3iT - 2.86e7T^{2} \)
37 \( 1 - 1.38e4iT - 6.93e7T^{2} \)
41 \( 1 - 1.35e4iT - 1.15e8T^{2} \)
43 \( 1 - 1.45e4T + 1.47e8T^{2} \)
47 \( 1 - 1.60e4iT - 2.29e8T^{2} \)
53 \( 1 + 8.22e3T + 4.18e8T^{2} \)
59 \( 1 - 7.47e3T + 7.14e8T^{2} \)
61 \( 1 + 1.23e4T + 8.44e8T^{2} \)
67 \( 1 + 6.06e4T + 1.35e9T^{2} \)
71 \( 1 - 2.83e4T + 1.80e9T^{2} \)
73 \( 1 + 4.02e4iT - 2.07e9T^{2} \)
79 \( 1 + 2.46e4iT - 3.07e9T^{2} \)
83 \( 1 + 2.54e4iT - 3.93e9T^{2} \)
89 \( 1 + 9.57e4iT - 5.58e9T^{2} \)
97 \( 1 - 1.34e5iT - 8.58e9T^{2} \)
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

−13.72894657955766635308268378269, −13.02171046945567702384644462367, −11.52411324781119034303456882964, −10.49092323109109617191054400145, −9.302954193237659586794189014594, −7.945203164579927824253906980610, −6.21325160446440011835338579040, −4.50555676303724620855183827185, −2.71084422868621848848533790846, −1.64180004357048091650605314300, 2.22018038344969400293853062376, 4.35267790266141600419900736079, 5.25716354618934864355282764829, 7.15864901472829378219094308358, 8.512412043312736174645449149250, 9.097276654870770792447686910189, 10.88324264141995143486742257678, 12.59480480777567997226781253159, 13.66399449072967972525258145760, 14.22789811796419498219771114818

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