Properties

Label 2-60-20.19-c4-0-20
Degree $2$
Conductor $60$
Sign $0.223 + 0.974i$
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 + (59.5 − 80.3i)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 + (0.595 − 0.803i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.223 + 0.974i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.223 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.223 + 0.974i$
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: \((2,\ 60,\ (\ :2),\ 0.223 + 0.974i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.21378 - 1.76378i\)
\(L(\frac12)\) \(\approx\) \(2.21378 - 1.76378i\)
\(L(3)\) 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

−14.01545903577442183685738578907, −13.00239787320851462393156113543, −12.15342981159064792668947160993, −10.35264121644270113738464697972, −9.890392165419894824446724510234, −8.332137317368840689901676394445, −6.32109909553416209626859538787, −4.97878507223035092709045434475, −3.16917918211986729886233169591, −1.64003478906768639400780554601, 2.49409499237916153630550242844, 4.34144011834822206339738024819, 5.93082334474081651358404149023, 7.07885686273009328094524835781, 8.604225459436306438524704682729, 9.560195844311936897802085758402, 11.35157613431989012728428626318, 12.79314707617215708499990113448, 13.85112523815134901573478500233, 14.23041282174013914708474101081

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