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Properties

Label 2-60-60.47-c2-0-11
Degree $2$
Conductor $60$
Sign $0.662 - 0.749i$
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  + (0.770 + 1.84i)2-s + (2.78 − 1.12i)3-s + (−2.81 + 2.84i)4-s + (3.86 − 3.17i)5-s + (4.21 + 4.26i)6-s + (−4.75 + 4.75i)7-s + (−7.41 − 2.99i)8-s + (6.46 − 6.25i)9-s + (8.83 + 4.68i)10-s − 11.9·11-s + (−4.61 + 11.0i)12-s + (−4.22 + 4.22i)13-s + (−12.4 − 5.10i)14-s + (7.16 − 13.1i)15-s + (−0.188 − 15.9i)16-s + (9.35 − 9.35i)17-s + ⋯
L(s)  = 1  + (0.385 + 0.922i)2-s + (0.927 − 0.375i)3-s + (−0.702 + 0.711i)4-s + (0.772 − 0.635i)5-s + (0.703 + 0.710i)6-s + (−0.678 + 0.678i)7-s + (−0.927 − 0.374i)8-s + (0.718 − 0.695i)9-s + (0.883 + 0.468i)10-s − 1.08·11-s + (−0.384 + 0.922i)12-s + (−0.324 + 0.324i)13-s + (−0.887 − 0.364i)14-s + (0.477 − 0.878i)15-s + (−0.0117 − 0.999i)16-s + (0.550 − 0.550i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.662 - 0.749i$
Analytic conductor: \(1.63488\)
Root analytic conductor: \(1.27862\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :1),\ 0.662 - 0.749i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.55211 + 0.699245i\)
\(L(\frac12)\) \(\approx\) \(1.55211 + 0.699245i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.770 - 1.84i)T \)
3 \( 1 + (-2.78 + 1.12i)T \)
5 \( 1 + (-3.86 + 3.17i)T \)
good7 \( 1 + (4.75 - 4.75i)T - 49iT^{2} \)
11 \( 1 + 11.9T + 121T^{2} \)
13 \( 1 + (4.22 - 4.22i)T - 169iT^{2} \)
17 \( 1 + (-9.35 + 9.35i)T - 289iT^{2} \)
19 \( 1 + 1.48T + 361T^{2} \)
23 \( 1 + (-11.6 + 11.6i)T - 529iT^{2} \)
29 \( 1 + 39.3T + 841T^{2} \)
31 \( 1 - 43.6iT - 961T^{2} \)
37 \( 1 + (-49.1 - 49.1i)T + 1.36e3iT^{2} \)
41 \( 1 - 27.0iT - 1.68e3T^{2} \)
43 \( 1 + (8.84 + 8.84i)T + 1.84e3iT^{2} \)
47 \( 1 + (-15.0 - 15.0i)T + 2.20e3iT^{2} \)
53 \( 1 + (-14.6 - 14.6i)T + 2.80e3iT^{2} \)
59 \( 1 + 61.7iT - 3.48e3T^{2} \)
61 \( 1 + 84.6T + 3.72e3T^{2} \)
67 \( 1 + (-65.7 + 65.7i)T - 4.48e3iT^{2} \)
71 \( 1 - 14.2T + 5.04e3T^{2} \)
73 \( 1 + (-16.0 + 16.0i)T - 5.32e3iT^{2} \)
79 \( 1 + 9.32T + 6.24e3T^{2} \)
83 \( 1 + (-12.7 + 12.7i)T - 6.88e3iT^{2} \)
89 \( 1 - 52.4T + 7.92e3T^{2} \)
97 \( 1 + (-6.90 - 6.90i)T + 9.40e3iT^{2} \)
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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

−14.94465693493712394755868223826, −13.87029232947481550188133524541, −13.00198727548046327717814048586, −12.37883659063933909750882776935, −9.752132340563835641059662601230, −8.963816189680548636803688616402, −7.79949538152231310319353448823, −6.38983987293409097075978838908, −5.00675068092397787636334906845, −2.86554707661891009993783022230, 2.45318139614105284025936500370, 3.73453285080929848733622886148, 5.59594390081347233620937610982, 7.56268358070826114832942240285, 9.362978718174466593050153337185, 10.12926189719952944655498088728, 10.90499444980353251211087869188, 12.99459827759945593498089379914, 13.33975000797715636432834181915, 14.51450235425009000257686733820

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