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Properties

Label 2-60-4.3-c2-0-2
Degree $2$
Conductor $60$
Sign $0.652 - 0.757i$
Analytic cond. $1.63488$
Root an. cond. $1.27862$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.87 + 0.696i)2-s + 1.73i·3-s + (3.02 + 2.61i)4-s − 2.23·5-s + (−1.20 + 3.24i)6-s − 5.46i·7-s + (3.86 + 7.00i)8-s − 2.99·9-s + (−4.19 − 1.55i)10-s − 11.0i·11-s + (−4.52 + 5.24i)12-s + 10.1·13-s + (3.80 − 10.2i)14-s − 3.87i·15-s + (2.35 + 15.8i)16-s − 24.4·17-s + ⋯
L(s)  = 1  + (0.937 + 0.348i)2-s + 0.577i·3-s + (0.757 + 0.652i)4-s − 0.447·5-s + (−0.201 + 0.541i)6-s − 0.781i·7-s + (0.482 + 0.875i)8-s − 0.333·9-s + (−0.419 − 0.155i)10-s − 1.00i·11-s + (−0.376 + 0.437i)12-s + 0.778·13-s + (0.272 − 0.732i)14-s − 0.258i·15-s + (0.147 + 0.989i)16-s − 1.43·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.652 - 0.757i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.652 - 0.757i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.652 - 0.757i$
Analytic conductor: \(1.63488\)
Root analytic conductor: \(1.27862\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :1),\ 0.652 - 0.757i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.64751 + 0.754894i\)
\(L(\frac12)\) \(\approx\) \(1.64751 + 0.754894i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.87 - 0.696i)T \)
3 \( 1 - 1.73iT \)
5 \( 1 + 2.23T \)
good7 \( 1 + 5.46iT - 49T^{2} \)
11 \( 1 + 11.0iT - 121T^{2} \)
13 \( 1 - 10.1T + 169T^{2} \)
17 \( 1 + 24.4T + 289T^{2} \)
19 \( 1 + 23.7iT - 361T^{2} \)
23 \( 1 - 37.2iT - 529T^{2} \)
29 \( 1 + 25.7T + 841T^{2} \)
31 \( 1 - 4.83iT - 961T^{2} \)
37 \( 1 - 35.6T + 1.36e3T^{2} \)
41 \( 1 + 9.30T + 1.68e3T^{2} \)
43 \( 1 - 70.0iT - 1.84e3T^{2} \)
47 \( 1 - 38.0iT - 2.20e3T^{2} \)
53 \( 1 - 55.7T + 2.80e3T^{2} \)
59 \( 1 + 55.5iT - 3.48e3T^{2} \)
61 \( 1 + 82.2T + 3.72e3T^{2} \)
67 \( 1 + 104. iT - 4.48e3T^{2} \)
71 \( 1 + 76.7iT - 5.04e3T^{2} \)
73 \( 1 + 93.5T + 5.32e3T^{2} \)
79 \( 1 - 49.3iT - 6.24e3T^{2} \)
83 \( 1 + 72.3iT - 6.88e3T^{2} \)
89 \( 1 - 115.T + 7.92e3T^{2} \)
97 \( 1 + 72.9T + 9.40e3T^{2} \)
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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

−15.14266159208055692536895638538, −13.74198850366734153139721119505, −13.21418607092761521164395937719, −11.32194690258666229098506841420, −11.02619108191801015359401944781, −8.979319572728341429896768485459, −7.58671423070604708341640069238, −6.18284594419800683595888343633, −4.58338015130897771512940328722, −3.39176709850731739691562024849, 2.21938489125390213593444402962, 4.20425441362113287206511896467, 5.87188499552584318229149665506, 7.09169350644890217384080884128, 8.705868342075748776036880166489, 10.45349238188677958767914772502, 11.68387788829845371377204892251, 12.47349256736219200160924257106, 13.36262251679893902362678248485, 14.70317771904347338427639483273

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