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Properties

Label 2-60-4.3-c6-0-15
Degree $2$
Conductor $60$
Sign $0.432 + 0.901i$
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  + (−7.80 + 1.77i)2-s + 15.5i·3-s + (57.7 − 27.6i)4-s + 55.9·5-s + (−27.6 − 121. i)6-s − 264. i·7-s + (−400. + 318. i)8-s − 243·9-s + (−436. + 99.1i)10-s + 29.4i·11-s + (431. + 899. i)12-s − 2.75e3·13-s + (469. + 2.06e3i)14-s + 871. i·15-s + (2.56e3 − 3.19e3i)16-s + 1.56e3·17-s + ⋯
L(s)  = 1  + (−0.975 + 0.221i)2-s + 0.577i·3-s + (0.901 − 0.432i)4-s + 0.447·5-s + (−0.128 − 0.562i)6-s − 0.771i·7-s + (−0.783 + 0.621i)8-s − 0.333·9-s + (−0.436 + 0.0991i)10-s + 0.0221i·11-s + (0.249 + 0.520i)12-s − 1.25·13-s + (0.171 + 0.751i)14-s + 0.258i·15-s + (0.625 − 0.780i)16-s + 0.317·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.432 + 0.901i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.432 + 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.432 + 0.901i$
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: \((2,\ 60,\ (\ :3),\ 0.432 + 0.901i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.726528 - 0.457244i\)
\(L(\frac12)\) \(\approx\) \(0.726528 - 0.457244i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (7.80 - 1.77i)T \)
3 \( 1 - 15.5iT \)
5 \( 1 - 55.9T \)
good7 \( 1 + 264. iT - 1.17e5T^{2} \)
11 \( 1 - 29.4iT - 1.77e6T^{2} \)
13 \( 1 + 2.75e3T + 4.82e6T^{2} \)
17 \( 1 - 1.56e3T + 2.41e7T^{2} \)
19 \( 1 + 7.29e3iT - 4.70e7T^{2} \)
23 \( 1 + 1.68e4iT - 1.48e8T^{2} \)
29 \( 1 - 3.62e4T + 5.94e8T^{2} \)
31 \( 1 + 4.54e4iT - 8.87e8T^{2} \)
37 \( 1 + 4.05e4T + 2.56e9T^{2} \)
41 \( 1 - 5.46e4T + 4.75e9T^{2} \)
43 \( 1 + 5.39e4iT - 6.32e9T^{2} \)
47 \( 1 - 1.12e5iT - 1.07e10T^{2} \)
53 \( 1 + 7.81e4T + 2.21e10T^{2} \)
59 \( 1 + 2.67e5iT - 4.21e10T^{2} \)
61 \( 1 + 3.53e5T + 5.15e10T^{2} \)
67 \( 1 + 2.03e5iT - 9.04e10T^{2} \)
71 \( 1 - 5.16e4iT - 1.28e11T^{2} \)
73 \( 1 - 2.21e5T + 1.51e11T^{2} \)
79 \( 1 + 5.40e5iT - 2.43e11T^{2} \)
83 \( 1 + 2.64e5iT - 3.26e11T^{2} \)
89 \( 1 - 3.14e5T + 4.96e11T^{2} \)
97 \( 1 + 1.55e5T + 8.32e11T^{2} \)
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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

−13.97901402683787106839901431564, −12.28166847023247001613682101750, −10.87601924437612846011975171218, −10.07279875229413787406898918782, −9.127665566462310915716965104164, −7.70634599545083185000034884444, −6.46289714233821027569104237902, −4.79339796521800687730408211153, −2.56369329099459605065160762300, −0.48818548313816948172583399565, 1.50466470088990049116118040383, 2.84893829858162252134673740662, 5.60852717269921820429124070074, 6.98925535738640807495658331700, 8.179068631794185601783266818295, 9.362576881822827283725771734835, 10.39300763150273012172186350024, 11.94969235612368286941425618371, 12.44362274806837738002518732225, 14.05853752728203152981884769186

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