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Properties

Label 2-60-20.3-c3-0-7
Degree $2$
Conductor $60$
Sign $0.707 - 0.706i$
Analytic cond. $3.54011$
Root an. cond. $1.88151$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.83 + 2.15i)2-s + (2.12 − 2.12i)3-s + (−1.25 + 7.90i)4-s + (8.47 − 7.28i)5-s + (8.45 + 0.665i)6-s + (13.7 + 13.7i)7-s + (−19.2 + 11.8i)8-s − 8.99i·9-s + (31.2 + 4.84i)10-s + 25.6i·11-s + (14.1 + 19.4i)12-s + (−46.9 − 46.9i)13-s + (−4.32 + 54.8i)14-s + (2.52 − 33.4i)15-s + (−60.8 − 19.7i)16-s + (38.6 − 38.6i)17-s + ⋯
L(s)  = 1  + (0.649 + 0.760i)2-s + (0.408 − 0.408i)3-s + (−0.156 + 0.987i)4-s + (0.758 − 0.651i)5-s + (0.575 + 0.0453i)6-s + (0.743 + 0.743i)7-s + (−0.852 + 0.522i)8-s − 0.333i·9-s + (0.988 + 0.153i)10-s + 0.702i·11-s + (0.339 + 0.467i)12-s + (−1.00 − 1.00i)13-s + (−0.0824 + 1.04i)14-s + (0.0434 − 0.575i)15-s + (−0.951 − 0.309i)16-s + (0.552 − 0.552i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.706i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.707 - 0.706i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.707 - 0.706i$
Analytic conductor: \(3.54011\)
Root analytic conductor: \(1.88151\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :3/2),\ 0.707 - 0.706i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.14734 + 0.888752i\)
\(L(\frac12)\) \(\approx\) \(2.14734 + 0.888752i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.83 - 2.15i)T \)
3 \( 1 + (-2.12 + 2.12i)T \)
5 \( 1 + (-8.47 + 7.28i)T \)
good7 \( 1 + (-13.7 - 13.7i)T + 343iT^{2} \)
11 \( 1 - 25.6iT - 1.33e3T^{2} \)
13 \( 1 + (46.9 + 46.9i)T + 2.19e3iT^{2} \)
17 \( 1 + (-38.6 + 38.6i)T - 4.91e3iT^{2} \)
19 \( 1 + 59.7T + 6.85e3T^{2} \)
23 \( 1 + (32.6 - 32.6i)T - 1.21e4iT^{2} \)
29 \( 1 + 28.5iT - 2.43e4T^{2} \)
31 \( 1 + 274. iT - 2.97e4T^{2} \)
37 \( 1 + (291. - 291. i)T - 5.06e4iT^{2} \)
41 \( 1 - 10.0T + 6.89e4T^{2} \)
43 \( 1 + (172. - 172. i)T - 7.95e4iT^{2} \)
47 \( 1 + (-430. - 430. i)T + 1.03e5iT^{2} \)
53 \( 1 + (-420. - 420. i)T + 1.48e5iT^{2} \)
59 \( 1 + 149.T + 2.05e5T^{2} \)
61 \( 1 - 394.T + 2.26e5T^{2} \)
67 \( 1 + (428. + 428. i)T + 3.00e5iT^{2} \)
71 \( 1 - 560. iT - 3.57e5T^{2} \)
73 \( 1 + (-480. - 480. i)T + 3.89e5iT^{2} \)
79 \( 1 - 347.T + 4.93e5T^{2} \)
83 \( 1 + (-217. + 217. i)T - 5.71e5iT^{2} \)
89 \( 1 + 1.44e3iT - 7.04e5T^{2} \)
97 \( 1 + (62.1 - 62.1i)T - 9.12e5iT^{2} \)
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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

−14.75213124016686709512917466777, −13.69387967343574825318730811432, −12.66076965340155753726146675118, −11.95870846748217311778295404415, −9.750170191104108230924335285491, −8.504563158472070453229451337155, −7.50297690214132316257347106641, −5.83683364696114725788055734520, −4.78090296596426730339108929374, −2.41009537531150444807167933230, 2.03303336169210542215346453516, 3.76267302056248615869971686047, 5.24562642966658231386244323035, 6.89595430520346569334142418584, 8.869208820143525543276724678841, 10.24798722319141981232231135348, 10.79065143257264985797993132170, 12.17047703911311730458989172491, 13.69487971197335307989610282817, 14.23954480212161116242926338786

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