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Properties

Label 2-60-20.7-c3-0-4
Degree $2$
Conductor $60$
Sign $-0.349 - 0.936i$
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  + (−2.82 − 0.129i)2-s + (2.12 + 2.12i)3-s + (7.96 + 0.731i)4-s + (−8.68 + 7.03i)5-s + (−5.71 − 6.26i)6-s + (4.43 − 4.43i)7-s + (−22.4 − 3.09i)8-s + 8.99i·9-s + (25.4 − 18.7i)10-s + 62.0i·11-s + (15.3 + 18.4i)12-s + (−59.9 + 59.9i)13-s + (−13.0 + 11.9i)14-s + (−33.3 − 3.49i)15-s + (62.9 + 11.6i)16-s + (−29.9 − 29.9i)17-s + ⋯
L(s)  = 1  + (−0.998 − 0.0457i)2-s + (0.408 + 0.408i)3-s + (0.995 + 0.0914i)4-s + (−0.776 + 0.629i)5-s + (−0.389 − 0.426i)6-s + (0.239 − 0.239i)7-s + (−0.990 − 0.136i)8-s + 0.333i·9-s + (0.804 − 0.593i)10-s + 1.69i·11-s + (0.369 + 0.443i)12-s + (−1.27 + 1.27i)13-s + (−0.249 + 0.228i)14-s + (−0.574 − 0.0602i)15-s + (0.983 + 0.182i)16-s + (−0.427 − 0.427i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.349 - 0.936i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.349 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $-0.349 - 0.936i$
Analytic conductor: \(3.54011\)
Root analytic conductor: \(1.88151\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :3/2),\ -0.349 - 0.936i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.431937 + 0.622458i\)
\(L(\frac12)\) \(\approx\) \(0.431937 + 0.622458i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.82 + 0.129i)T \)
3 \( 1 + (-2.12 - 2.12i)T \)
5 \( 1 + (8.68 - 7.03i)T \)
good7 \( 1 + (-4.43 + 4.43i)T - 343iT^{2} \)
11 \( 1 - 62.0iT - 1.33e3T^{2} \)
13 \( 1 + (59.9 - 59.9i)T - 2.19e3iT^{2} \)
17 \( 1 + (29.9 + 29.9i)T + 4.91e3iT^{2} \)
19 \( 1 - 94.9T + 6.85e3T^{2} \)
23 \( 1 + (-16.3 - 16.3i)T + 1.21e4iT^{2} \)
29 \( 1 + 129. iT - 2.43e4T^{2} \)
31 \( 1 - 15.9iT - 2.97e4T^{2} \)
37 \( 1 + (-67.1 - 67.1i)T + 5.06e4iT^{2} \)
41 \( 1 - 368.T + 6.89e4T^{2} \)
43 \( 1 + (-10.5 - 10.5i)T + 7.95e4iT^{2} \)
47 \( 1 + (-279. + 279. i)T - 1.03e5iT^{2} \)
53 \( 1 + (19.3 - 19.3i)T - 1.48e5iT^{2} \)
59 \( 1 - 282.T + 2.05e5T^{2} \)
61 \( 1 + 316.T + 2.26e5T^{2} \)
67 \( 1 + (60.2 - 60.2i)T - 3.00e5iT^{2} \)
71 \( 1 - 316. iT - 3.57e5T^{2} \)
73 \( 1 + (-13.9 + 13.9i)T - 3.89e5iT^{2} \)
79 \( 1 + 837.T + 4.93e5T^{2} \)
83 \( 1 + (-405. - 405. i)T + 5.71e5iT^{2} \)
89 \( 1 + 340. iT - 7.04e5T^{2} \)
97 \( 1 + (-308. - 308. i)T + 9.12e5iT^{2} \)
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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

−15.10671014700277469549978125446, −14.27565157045915314627504971492, −12.20787351535384804556718817304, −11.39981241320907995425336561194, −10.04990955071692238103646680788, −9.300245411269056357934735821022, −7.59880739381324702348696970492, −7.07005860684675323054001751796, −4.41271060016707716334171751357, −2.41732118192754718301029986992, 0.69353241088429805385178941198, 3.04696710031761333359569328032, 5.60913935855644704565286789814, 7.46101375382597311892735020930, 8.252244309235105017766192260776, 9.204896598990415677754029793778, 10.79243027815024888548273704114, 11.86541555284089971868508665976, 12.88299512843153924457859528479, 14.49308072991943493247065845303

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