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Properties

Label 2-60-4.3-c6-0-17
Degree $2$
Conductor $60$
Sign $0.776 + 0.630i$
Analytic cond. $13.8032$
Root an. cond. $3.71527$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.43 + 7.22i)2-s − 15.5i·3-s + (−40.3 + 49.6i)4-s + 55.9·5-s + (112. − 53.6i)6-s − 472. i·7-s + (−497. − 120. i)8-s − 243·9-s + (192. + 403. i)10-s − 939. i·11-s + (774. + 628. i)12-s + 22.5·13-s + (3.40e3 − 1.62e3i)14-s − 871. i·15-s + (−839. − 4.00e3i)16-s + 6.24e3·17-s + ⋯
L(s)  = 1  + (0.429 + 0.902i)2-s − 0.577i·3-s + (−0.630 + 0.776i)4-s + 0.447·5-s + (0.521 − 0.248i)6-s − 1.37i·7-s + (−0.971 − 0.235i)8-s − 0.333·9-s + (0.192 + 0.403i)10-s − 0.705i·11-s + (0.448 + 0.363i)12-s + 0.0102·13-s + (1.24 − 0.591i)14-s − 0.258i·15-s + (−0.205 − 0.978i)16-s + 1.27·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.776 + 0.630i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.776 + 0.630i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.776 + 0.630i$
Analytic conductor: \(13.8032\)
Root analytic conductor: \(3.71527\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :3),\ 0.776 + 0.630i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.80964 - 0.642324i\)
\(L(\frac12)\) \(\approx\) \(1.80964 - 0.642324i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-3.43 - 7.22i)T \)
3 \( 1 + 15.5iT \)
5 \( 1 - 55.9T \)
good7 \( 1 + 472. iT - 1.17e5T^{2} \)
11 \( 1 + 939. iT - 1.77e6T^{2} \)
13 \( 1 - 22.5T + 4.82e6T^{2} \)
17 \( 1 - 6.24e3T + 2.41e7T^{2} \)
19 \( 1 + 1.11e4iT - 4.70e7T^{2} \)
23 \( 1 - 1.16e3iT - 1.48e8T^{2} \)
29 \( 1 - 2.56e4T + 5.94e8T^{2} \)
31 \( 1 + 4.30e3iT - 8.87e8T^{2} \)
37 \( 1 + 5.06e4T + 2.56e9T^{2} \)
41 \( 1 + 1.13e5T + 4.75e9T^{2} \)
43 \( 1 - 1.13e5iT - 6.32e9T^{2} \)
47 \( 1 + 1.84e5iT - 1.07e10T^{2} \)
53 \( 1 + 1.33e5T + 2.21e10T^{2} \)
59 \( 1 - 5.52e4iT - 4.21e10T^{2} \)
61 \( 1 - 2.04e5T + 5.15e10T^{2} \)
67 \( 1 - 2.44e5iT - 9.04e10T^{2} \)
71 \( 1 - 1.42e4iT - 1.28e11T^{2} \)
73 \( 1 + 2.78e5T + 1.51e11T^{2} \)
79 \( 1 - 3.58e5iT - 2.43e11T^{2} \)
83 \( 1 + 8.20e5iT - 3.26e11T^{2} \)
89 \( 1 - 1.00e6T + 4.96e11T^{2} \)
97 \( 1 - 1.44e6T + 8.32e11T^{2} \)
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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

−13.69690435970991266512571273285, −13.14952088256540945995952024829, −11.69992112362569111733611813325, −10.16508264499657744164560524598, −8.612763379477915532308966759441, −7.39836901089752956905806442014, −6.47841565912879319067595890757, −5.01517771752437381037602501832, −3.31453496923119398827691350887, −0.71992672748863046168616175026, 1.83217303117431974256048513538, 3.29651183640177653815962245587, 5.02114508524689361026293989575, 5.97258133657949019999691542556, 8.478643829708453231638159923709, 9.659041027318759966652772307529, 10.39007017765060857821831286285, 12.00424550102071499249803264801, 12.42916923255318103132614999035, 14.05927006015055163858849270674

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