↑

Properties

Label 2-60-20.19-c6-0-34
Degree $2$
Conductor $60$
Sign $-0.285 + 0.958i$
Analytic cond. $13.8032$
Root an. cond. $3.71527$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (7.34 − 3.15i)2-s − 15.5·3-s + (44.0 − 46.4i)4-s + (111. − 56.4i)5-s + (−114. + 49.2i)6-s − 231.·7-s + (176. − 480. i)8-s + 243·9-s + (641. − 767. i)10-s − 1.43e3i·11-s + (−686. + 723. i)12-s + 1.70e3i·13-s + (−1.70e3 + 731. i)14-s + (−1.73e3 + 880. i)15-s + (−217. − 4.09e3i)16-s − 7.56e3i·17-s + ⋯
L(s)  = 1  + (0.918 − 0.394i)2-s − 0.577·3-s + (0.688 − 0.725i)4-s + (0.892 − 0.451i)5-s + (−0.530 + 0.228i)6-s − 0.675·7-s + (0.345 − 0.938i)8-s + 0.333·9-s + (0.641 − 0.767i)10-s − 1.07i·11-s + (−0.397 + 0.418i)12-s + 0.775i·13-s + (−0.620 + 0.266i)14-s + (−0.515 + 0.260i)15-s + (−0.0531 − 0.998i)16-s − 1.54i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.285 + 0.958i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.285 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $-0.285 + 0.958i$
Analytic conductor: \(13.8032\)
Root analytic conductor: \(3.71527\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :3),\ -0.285 + 0.958i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.57979 - 2.11989i\)
\(L(\frac12)\) \(\approx\) \(1.57979 - 2.11989i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-7.34 + 3.15i)T \)
3 \( 1 + 15.5T \)
5 \( 1 + (-111. + 56.4i)T \)
good7 \( 1 + 231.T + 1.17e5T^{2} \)
11 \( 1 + 1.43e3iT - 1.77e6T^{2} \)
13 \( 1 - 1.70e3iT - 4.82e6T^{2} \)
17 \( 1 + 7.56e3iT - 2.41e7T^{2} \)
19 \( 1 + 5.56e3iT - 4.70e7T^{2} \)
23 \( 1 + 1.44e4T + 1.48e8T^{2} \)
29 \( 1 - 2.57e4T + 5.94e8T^{2} \)
31 \( 1 - 4.09e4iT - 8.87e8T^{2} \)
37 \( 1 - 6.60e4iT - 2.56e9T^{2} \)
41 \( 1 - 1.06e5T + 4.75e9T^{2} \)
43 \( 1 - 5.83e4T + 6.32e9T^{2} \)
47 \( 1 - 7.50e4T + 1.07e10T^{2} \)
53 \( 1 - 8.18e4iT - 2.21e10T^{2} \)
59 \( 1 - 1.99e5iT - 4.21e10T^{2} \)
61 \( 1 - 1.68e5T + 5.15e10T^{2} \)
67 \( 1 + 3.68e4T + 9.04e10T^{2} \)
71 \( 1 - 4.49e5iT - 1.28e11T^{2} \)
73 \( 1 + 5.85e4iT - 1.51e11T^{2} \)
79 \( 1 + 6.49e5iT - 2.43e11T^{2} \)
83 \( 1 - 4.52e5T + 3.26e11T^{2} \)
89 \( 1 + 2.86e5T + 4.96e11T^{2} \)
97 \( 1 + 3.86e5iT - 8.32e11T^{2} \)
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

−13.61338815352871475616406642130, −12.41666928186333616055636310058, −11.46772063550448388908820868537, −10.23528226447418464387700621834, −9.163933289214027253125255229359, −6.78056821933558840374317524390, −5.84124015276142583221124702233, −4.63198976972210415008084251931, −2.75435211291288836058281385365, −0.884972076453369723481849813158, 2.17074144567863165392357791795, 3.97887760362315172644146636183, 5.70264270546215930047442782062, 6.38120855019660490273900074422, 7.77521015551838406222805160369, 9.853853165113877958135155016910, 10.77198292963936799138862512909, 12.41757950718621526977271971076, 12.89423068380141644061967431649, 14.21537997725702580151106133894

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