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Properties

Label 2-60-20.19-c4-0-6
Degree $2$
Conductor $60$
Sign $-0.0872 - 0.996i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.59 + 3.04i)2-s − 5.19·3-s + (−2.49 − 15.8i)4-s + (24.9 + 1.72i)5-s + (13.5 − 15.7i)6-s − 2.65·7-s + (54.5 + 33.5i)8-s + 27·9-s + (−70.0 + 71.3i)10-s + 60.4i·11-s + (12.9 + 82.1i)12-s + 211. i·13-s + (6.88 − 8.05i)14-s + (−129. − 8.95i)15-s + (−243. + 78.7i)16-s − 10.2i·17-s + ⋯
L(s)  = 1  + (−0.649 + 0.760i)2-s − 0.577·3-s + (−0.155 − 0.987i)4-s + (0.997 + 0.0689i)5-s + (0.375 − 0.438i)6-s − 0.0540·7-s + (0.852 + 0.523i)8-s + 0.333·9-s + (−0.700 + 0.713i)10-s + 0.499i·11-s + (0.0898 + 0.570i)12-s + 1.24i·13-s + (0.0351 − 0.0411i)14-s + (−0.575 − 0.0398i)15-s + (−0.951 + 0.307i)16-s − 0.0355i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0872 - 0.996i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.0872 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $-0.0872 - 0.996i$
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: \((2,\ 60,\ (\ :2),\ -0.0872 - 0.996i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.693230 + 0.756567i\)
\(L(\frac12)\) \(\approx\) \(0.693230 + 0.756567i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.59 - 3.04i)T \)
3 \( 1 + 5.19T \)
5 \( 1 + (-24.9 - 1.72i)T \)
good7 \( 1 + 2.65T + 2.40e3T^{2} \)
11 \( 1 - 60.4iT - 1.46e4T^{2} \)
13 \( 1 - 211. iT - 2.85e4T^{2} \)
17 \( 1 + 10.2iT - 8.35e4T^{2} \)
19 \( 1 - 484. iT - 1.30e5T^{2} \)
23 \( 1 - 558.T + 2.79e5T^{2} \)
29 \( 1 - 948.T + 7.07e5T^{2} \)
31 \( 1 - 1.40e3iT - 9.23e5T^{2} \)
37 \( 1 + 2.31e3iT - 1.87e6T^{2} \)
41 \( 1 + 585.T + 2.82e6T^{2} \)
43 \( 1 + 1.02e3T + 3.41e6T^{2} \)
47 \( 1 - 940.T + 4.87e6T^{2} \)
53 \( 1 - 4.05e3iT - 7.89e6T^{2} \)
59 \( 1 + 2.58e3iT - 1.21e7T^{2} \)
61 \( 1 - 174.T + 1.38e7T^{2} \)
67 \( 1 + 42.9T + 2.01e7T^{2} \)
71 \( 1 + 4.33e3iT - 2.54e7T^{2} \)
73 \( 1 - 3.22e3iT - 2.83e7T^{2} \)
79 \( 1 + 1.08e4iT - 3.89e7T^{2} \)
83 \( 1 - 9.39e3T + 4.74e7T^{2} \)
89 \( 1 + 1.15e4T + 6.27e7T^{2} \)
97 \( 1 + 1.37e4iT - 8.85e7T^{2} \)
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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

−14.62583584044477844541352490696, −13.85489859351726970066804543768, −12.39160839006506883565519980615, −10.83205391541810257423895739143, −9.879501492194410051080742311724, −8.862465514570497320302419428249, −7.11631402843229686758532077481, −6.16334355744033024903252132930, −4.87506795578580003175915363824, −1.60074731182959049733638684858, 0.846643384143304067656963787381, 2.83477130255971993982721583639, 5.05180586085530117385166009855, 6.66561167174189468231815739339, 8.342038544743521886066319185617, 9.617811751863822455442300665927, 10.54656596328770450905762754312, 11.52455891944014808994446007992, 12.92063703689931548968144918226, 13.49121667877178592767229245087

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