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Properties

Label 2-60-60.47-c2-0-4
Degree $2$
Conductor $60$
Sign $-0.157 - 0.987i$
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  + (−1.75 + 0.961i)2-s + (0.903 + 2.86i)3-s + (2.15 − 3.37i)4-s + (4.95 + 0.663i)5-s + (−4.33 − 4.14i)6-s + (−7.30 + 7.30i)7-s + (−0.535 + 7.98i)8-s + (−7.36 + 5.17i)9-s + (−9.32 + 3.59i)10-s + 4.41·11-s + (11.5 + 3.11i)12-s + (7.53 − 7.53i)13-s + (5.78 − 19.8i)14-s + (2.58 + 14.7i)15-s + (−6.73 − 14.5i)16-s + (−0.350 + 0.350i)17-s + ⋯
L(s)  = 1  + (−0.876 + 0.480i)2-s + (0.301 + 0.953i)3-s + (0.538 − 0.842i)4-s + (0.991 + 0.132i)5-s + (−0.722 − 0.691i)6-s + (−1.04 + 1.04i)7-s + (−0.0669 + 0.997i)8-s + (−0.818 + 0.574i)9-s + (−0.932 + 0.359i)10-s + 0.401·11-s + (0.965 + 0.259i)12-s + (0.579 − 0.579i)13-s + (0.413 − 1.41i)14-s + (0.172 + 0.985i)15-s + (−0.420 − 0.907i)16-s + (−0.0206 + 0.0206i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.157 - 0.987i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.157 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $-0.157 - 0.987i$
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: \((2,\ 60,\ (\ :1),\ -0.157 - 0.987i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.602983 + 0.706939i\)
\(L(\frac12)\) \(\approx\) \(0.602983 + 0.706939i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.75 - 0.961i)T \)
3 \( 1 + (-0.903 - 2.86i)T \)
5 \( 1 + (-4.95 - 0.663i)T \)
good7 \( 1 + (7.30 - 7.30i)T - 49iT^{2} \)
11 \( 1 - 4.41T + 121T^{2} \)
13 \( 1 + (-7.53 + 7.53i)T - 169iT^{2} \)
17 \( 1 + (0.350 - 0.350i)T - 289iT^{2} \)
19 \( 1 - 9.24T + 361T^{2} \)
23 \( 1 + (-17.9 + 17.9i)T - 529iT^{2} \)
29 \( 1 - 5.52T + 841T^{2} \)
31 \( 1 + 48.1iT - 961T^{2} \)
37 \( 1 + (-3.39 - 3.39i)T + 1.36e3iT^{2} \)
41 \( 1 - 33.0iT - 1.68e3T^{2} \)
43 \( 1 + (-1.45 - 1.45i)T + 1.84e3iT^{2} \)
47 \( 1 + (27.8 + 27.8i)T + 2.20e3iT^{2} \)
53 \( 1 + (52.6 + 52.6i)T + 2.80e3iT^{2} \)
59 \( 1 - 24.6iT - 3.48e3T^{2} \)
61 \( 1 - 46.1T + 3.72e3T^{2} \)
67 \( 1 + (32.1 - 32.1i)T - 4.48e3iT^{2} \)
71 \( 1 - 116.T + 5.04e3T^{2} \)
73 \( 1 + (72.2 - 72.2i)T - 5.32e3iT^{2} \)
79 \( 1 + 55.9T + 6.24e3T^{2} \)
83 \( 1 + (46.5 - 46.5i)T - 6.88e3iT^{2} \)
89 \( 1 + 33.2T + 7.92e3T^{2} \)
97 \( 1 + (24.6 + 24.6i)T + 9.40e3iT^{2} \)
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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

−15.33015406464679778388060598886, −14.48253723762563881538272553663, −13.12441848010360755528814782641, −11.31128724924810814049185129729, −10.03785793334837352169504010770, −9.410127644563392754380851798174, −8.477347057836466198876170883750, −6.41705630496557290650920773404, −5.46122981099885893695941770774, −2.75504241564119711309187992230, 1.31585993617653015168059582112, 3.26485853351258955966322827421, 6.40296220975341106618860511670, 7.21986352445130882492419696760, 8.855164866139588679555367008291, 9.705764532318983992287685610718, 10.97715623158191874962860734231, 12.40243795602655018462656548002, 13.31575660838173715613978655490, 14.05656377703685320196852076223

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