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Properties

Label 2-60-60.47-c2-0-5
Degree $2$
Conductor $60$
Sign $-0.487 - 0.873i$
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  + (−0.141 + 1.99i)2-s + (2.06 + 2.17i)3-s + (−3.95 − 0.565i)4-s + (−3.07 + 3.94i)5-s + (−4.63 + 3.81i)6-s + (5.18 − 5.18i)7-s + (1.68 − 7.81i)8-s + (−0.459 + 8.98i)9-s + (−7.42 − 6.69i)10-s + 7.14·11-s + (−6.95 − 9.78i)12-s + (−7.93 + 7.93i)13-s + (9.61 + 11.0i)14-s + (−14.9 + 1.45i)15-s + (15.3 + 4.47i)16-s + (16.5 − 16.5i)17-s + ⋯
L(s)  = 1  + (−0.0708 + 0.997i)2-s + (0.688 + 0.724i)3-s + (−0.989 − 0.141i)4-s + (−0.615 + 0.788i)5-s + (−0.771 + 0.635i)6-s + (0.741 − 0.741i)7-s + (0.211 − 0.977i)8-s + (−0.0510 + 0.998i)9-s + (−0.742 − 0.669i)10-s + 0.649·11-s + (−0.579 − 0.815i)12-s + (−0.610 + 0.610i)13-s + (0.686 + 0.791i)14-s + (−0.995 + 0.0970i)15-s + (0.960 + 0.279i)16-s + (0.975 − 0.975i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.487 - 0.873i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.487 - 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $-0.487 - 0.873i$
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: \((2,\ 60,\ (\ :1),\ -0.487 - 0.873i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.627132 + 1.06805i\)
\(L(\frac12)\) \(\approx\) \(0.627132 + 1.06805i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.141 - 1.99i)T \)
3 \( 1 + (-2.06 - 2.17i)T \)
5 \( 1 + (3.07 - 3.94i)T \)
good7 \( 1 + (-5.18 + 5.18i)T - 49iT^{2} \)
11 \( 1 - 7.14T + 121T^{2} \)
13 \( 1 + (7.93 - 7.93i)T - 169iT^{2} \)
17 \( 1 + (-16.5 + 16.5i)T - 289iT^{2} \)
19 \( 1 - 12.1T + 361T^{2} \)
23 \( 1 + (-11.0 + 11.0i)T - 529iT^{2} \)
29 \( 1 + 26.1T + 841T^{2} \)
31 \( 1 + 8.74iT - 961T^{2} \)
37 \( 1 + (26.7 + 26.7i)T + 1.36e3iT^{2} \)
41 \( 1 - 35.4iT - 1.68e3T^{2} \)
43 \( 1 + (24.6 + 24.6i)T + 1.84e3iT^{2} \)
47 \( 1 + (58.6 + 58.6i)T + 2.20e3iT^{2} \)
53 \( 1 + (-20.4 - 20.4i)T + 2.80e3iT^{2} \)
59 \( 1 - 59.4iT - 3.48e3T^{2} \)
61 \( 1 - 7.42T + 3.72e3T^{2} \)
67 \( 1 + (-35.8 + 35.8i)T - 4.48e3iT^{2} \)
71 \( 1 + 46.2T + 5.04e3T^{2} \)
73 \( 1 + (-10.6 + 10.6i)T - 5.32e3iT^{2} \)
79 \( 1 - 68.3T + 6.24e3T^{2} \)
83 \( 1 + (-76.6 + 76.6i)T - 6.88e3iT^{2} \)
89 \( 1 + 41.0T + 7.92e3T^{2} \)
97 \( 1 + (-81.7 - 81.7i)T + 9.40e3iT^{2} \)
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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

−14.95171109182873527425516425568, −14.46220797159296828913254125055, −13.70565106353957756002580546016, −11.68372920558483051429109415017, −10.33374992423712664767335828921, −9.240135218792550567951502108255, −7.83923092605009359244435837325, −7.08622347256304446753216816322, −4.92573237933756719847504297073, −3.67595935969080502441104486883, 1.49424293576142110239934767359, 3.47352667329462063248270701246, 5.26310082951561988648122038188, 7.77442845090183699804159297294, 8.563648384889834435154105980298, 9.627350167327223396938052648253, 11.48853148277838538548343044870, 12.24322024973181737946501236716, 12.98920010558247729221380314942, 14.35435344268478036893999282016

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