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Properties

Label 2-60-20.7-c3-0-8
Degree $2$
Conductor $60$
Sign $0.926 - 0.376i$
Analytic cond. $3.54011$
Root an. cond. $1.88151$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.23 + 1.72i)2-s + (2.12 + 2.12i)3-s + (2.01 − 7.74i)4-s + (7.18 − 8.56i)5-s + (−8.41 − 1.07i)6-s + (13.0 − 13.0i)7-s + (8.87 + 20.8i)8-s + 8.99i·9-s + (−1.26 + 31.5i)10-s + 38.8i·11-s + (20.7 − 12.1i)12-s + (41.6 − 41.6i)13-s + (−6.62 + 51.7i)14-s + (33.4 − 2.93i)15-s + (−55.8 − 31.2i)16-s + (80.8 + 80.8i)17-s + ⋯
L(s)  = 1  + (−0.791 + 0.611i)2-s + (0.408 + 0.408i)3-s + (0.252 − 0.967i)4-s + (0.642 − 0.766i)5-s + (−0.572 − 0.0733i)6-s + (0.703 − 0.703i)7-s + (0.392 + 0.919i)8-s + 0.333i·9-s + (−0.0398 + 0.999i)10-s + 1.06i·11-s + (0.497 − 0.292i)12-s + (0.889 − 0.889i)13-s + (−0.126 + 0.987i)14-s + (0.575 − 0.0504i)15-s + (−0.872 − 0.487i)16-s + (1.15 + 1.15i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.926 - 0.376i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.926 - 0.376i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.926 - 0.376i$
Analytic conductor: \(3.54011\)
Root analytic conductor: \(1.88151\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :3/2),\ 0.926 - 0.376i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.28124 + 0.250561i\)
\(L(\frac12)\) \(\approx\) \(1.28124 + 0.250561i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.23 - 1.72i)T \)
3 \( 1 + (-2.12 - 2.12i)T \)
5 \( 1 + (-7.18 + 8.56i)T \)
good7 \( 1 + (-13.0 + 13.0i)T - 343iT^{2} \)
11 \( 1 - 38.8iT - 1.33e3T^{2} \)
13 \( 1 + (-41.6 + 41.6i)T - 2.19e3iT^{2} \)
17 \( 1 + (-80.8 - 80.8i)T + 4.91e3iT^{2} \)
19 \( 1 + 23.9T + 6.85e3T^{2} \)
23 \( 1 + (131. + 131. i)T + 1.21e4iT^{2} \)
29 \( 1 + 17.3iT - 2.43e4T^{2} \)
31 \( 1 - 120. iT - 2.97e4T^{2} \)
37 \( 1 + (-17.5 - 17.5i)T + 5.06e4iT^{2} \)
41 \( 1 + 228.T + 6.89e4T^{2} \)
43 \( 1 + (-64.3 - 64.3i)T + 7.95e4iT^{2} \)
47 \( 1 + (-49.7 + 49.7i)T - 1.03e5iT^{2} \)
53 \( 1 + (263. - 263. i)T - 1.48e5iT^{2} \)
59 \( 1 + 613.T + 2.05e5T^{2} \)
61 \( 1 + 406.T + 2.26e5T^{2} \)
67 \( 1 + (-246. + 246. i)T - 3.00e5iT^{2} \)
71 \( 1 - 226. iT - 3.57e5T^{2} \)
73 \( 1 + (652. - 652. i)T - 3.89e5iT^{2} \)
79 \( 1 + 308.T + 4.93e5T^{2} \)
83 \( 1 + (690. + 690. i)T + 5.71e5iT^{2} \)
89 \( 1 - 101. iT - 7.04e5T^{2} \)
97 \( 1 + (-921. - 921. i)T + 9.12e5iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.74976470439066346794592913549, −13.92410367728843369167991394046, −12.52813602905533068238936588936, −10.57204377385868380761127818793, −10.01707889508334187540103824728, −8.586860942321624769441199133442, −7.81619919422127294018262170770, −5.99132936858274591352216892543, −4.56046979739892821640658496538, −1.52045120469285043799946975115, 1.76486194802939626722346624912, 3.27920170406187181805384193738, 6.02880223169462295011257085714, 7.56760213602235489730598364085, 8.722590839682261277503517464911, 9.758577928242361975849224017563, 11.20202587589346233225224854079, 11.86821945697044363486227351864, 13.52919237475928740426113774016, 14.20283450398671997838647807511

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