↑

Properties

Label 2-60-20.19-c4-0-15
Degree $2$
Conductor $60$
Sign $0.925 - 0.378i$
Analytic cond. $6.20219$
Root an. cond. $2.49042$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (3.76 + 1.34i)2-s − 5.19·3-s + (12.4 + 10.1i)4-s + (11.9 − 21.9i)5-s + (−19.5 − 6.96i)6-s + 63.7·7-s + (33.2 + 54.6i)8-s + 27·9-s + (74.5 − 66.6i)10-s + 181. i·11-s + (−64.4 − 52.4i)12-s − 1.57i·13-s + (240. + 85.4i)14-s + (−62.2 + 114. i)15-s + (51.9 + 250. i)16-s − 483. i·17-s + ⋯
L(s)  = 1  + (0.942 + 0.335i)2-s − 0.577·3-s + (0.775 + 0.631i)4-s + (0.479 − 0.877i)5-s + (−0.543 − 0.193i)6-s + 1.30·7-s + (0.519 + 0.854i)8-s + 0.333·9-s + (0.745 − 0.666i)10-s + 1.50i·11-s + (−0.447 − 0.364i)12-s − 0.00931i·13-s + (1.22 + 0.436i)14-s + (−0.276 + 0.506i)15-s + (0.202 + 0.979i)16-s − 1.67i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.925 - 0.378i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.925 - 0.378i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.925 - 0.378i$
Analytic conductor: \(6.20219\)
Root analytic conductor: \(2.49042\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :2),\ 0.925 - 0.378i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.64880 + 0.520128i\)
\(L(\frac12)\) \(\approx\) \(2.64880 + 0.520128i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-3.76 - 1.34i)T \)
3 \( 1 + 5.19T \)
5 \( 1 + (-11.9 + 21.9i)T \)
good7 \( 1 - 63.7T + 2.40e3T^{2} \)
11 \( 1 - 181. iT - 1.46e4T^{2} \)
13 \( 1 + 1.57iT - 2.85e4T^{2} \)
17 \( 1 + 483. iT - 8.35e4T^{2} \)
19 \( 1 + 192. iT - 1.30e5T^{2} \)
23 \( 1 + 558.T + 2.79e5T^{2} \)
29 \( 1 - 51.9T + 7.07e5T^{2} \)
31 \( 1 - 310. iT - 9.23e5T^{2} \)
37 \( 1 - 1.46e3iT - 1.87e6T^{2} \)
41 \( 1 + 2.62e3T + 2.82e6T^{2} \)
43 \( 1 - 203.T + 3.41e6T^{2} \)
47 \( 1 + 3.73e3T + 4.87e6T^{2} \)
53 \( 1 + 684. iT - 7.89e6T^{2} \)
59 \( 1 + 2.36e3iT - 1.21e7T^{2} \)
61 \( 1 - 5.34e3T + 1.38e7T^{2} \)
67 \( 1 - 561.T + 2.01e7T^{2} \)
71 \( 1 + 5.99e3iT - 2.54e7T^{2} \)
73 \( 1 - 6.56e3iT - 2.83e7T^{2} \)
79 \( 1 - 1.37e3iT - 3.89e7T^{2} \)
83 \( 1 - 3.15e3T + 4.74e7T^{2} \)
89 \( 1 - 1.07e4T + 6.27e7T^{2} \)
97 \( 1 + 9.15e3iT - 8.85e7T^{2} \)
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

−14.32240331014323995639265442119, −13.31933383121143161298353094486, −12.14473526682891461768965476322, −11.50989534937455398956735471168, −9.868202142743246533404408210246, −8.138867611216596673296858751625, −6.86234688058866704481689020346, −5.11618724473351794444589668329, −4.68693099463808096084098822525, −1.87047466615932172717710424893, 1.76618806103459334110627579040, 3.75765658902409346764449286503, 5.50402837134088868309493783134, 6.34866133811451687808602323838, 8.067303065186526486376868751194, 10.29454195950465604460259479609, 11.00648791046369345827839205907, 11.81916508686335582230046066327, 13.27736163262260567529321424732, 14.27340180006935314295208918330

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