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Properties

Label 2-60-12.11-c1-0-3
Degree $2$
Conductor $60$
Sign $0.993 - 0.110i$
Analytic cond. $0.479102$
Root an. cond. $0.692172$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.17 + 0.780i)2-s + (1.51 − 0.848i)3-s + (0.780 − 1.84i)4-s − i·5-s + (−1.11 + 2.17i)6-s + 3.02i·7-s + (0.516 + 2.78i)8-s + (1.56 − 2.56i)9-s + (0.780 + 1.17i)10-s − 1.32·11-s + (−0.382 − 3.44i)12-s − 5.12·13-s + (−2.35 − 3.56i)14-s + (−0.848 − 1.51i)15-s + (−2.78 − 2.87i)16-s + 2i·17-s + ⋯
L(s)  = 1  + (−0.833 + 0.552i)2-s + (0.871 − 0.489i)3-s + (0.390 − 0.920i)4-s − 0.447i·5-s + (−0.456 + 0.889i)6-s + 1.14i·7-s + (0.182 + 0.983i)8-s + (0.520 − 0.853i)9-s + (0.246 + 0.372i)10-s − 0.399·11-s + (−0.110 − 0.993i)12-s − 1.42·13-s + (−0.630 − 0.951i)14-s + (−0.218 − 0.389i)15-s + (−0.695 − 0.718i)16-s + 0.485i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.110i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 - 0.110i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.993 - 0.110i$
Analytic conductor: \(0.479102\)
Root analytic conductor: \(0.692172\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :1/2),\ 0.993 - 0.110i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.758574 + 0.0419982i\)
\(L(\frac12)\) \(\approx\) \(0.758574 + 0.0419982i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.17 - 0.780i)T \)
3 \( 1 + (-1.51 + 0.848i)T \)
5 \( 1 + iT \)
good7 \( 1 - 3.02iT - 7T^{2} \)
11 \( 1 + 1.32T + 11T^{2} \)
13 \( 1 + 5.12T + 13T^{2} \)
17 \( 1 - 2iT - 17T^{2} \)
19 \( 1 + 1.32iT - 19T^{2} \)
23 \( 1 + 0.371T + 23T^{2} \)
29 \( 1 - 3.12iT - 29T^{2} \)
31 \( 1 - 4.71iT - 31T^{2} \)
37 \( 1 - 5.12T + 37T^{2} \)
41 \( 1 + 1.12iT - 41T^{2} \)
43 \( 1 + 7.73iT - 43T^{2} \)
47 \( 1 + 3.02T + 47T^{2} \)
53 \( 1 + 12.2iT - 53T^{2} \)
59 \( 1 - 14.1T + 59T^{2} \)
61 \( 1 - 3.12T + 61T^{2} \)
67 \( 1 - 4.34iT - 67T^{2} \)
71 \( 1 + 3.39T + 71T^{2} \)
73 \( 1 - 8.24T + 73T^{2} \)
79 \( 1 + 8.10iT - 79T^{2} \)
83 \( 1 + 15.1T + 83T^{2} \)
89 \( 1 - 10.2iT - 89T^{2} \)
97 \( 1 + 6T + 97T^{2} \)
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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

−15.12407479916955656824875434893, −14.42469899769261099193997714919, −12.90973305905867438056492609908, −11.87238853578059910031331998057, −10.02214145196781651648096060403, −9.005623193342057176763778508592, −8.166632937750527930853992492516, −6.92714077433312159736841806645, −5.30164997060275174402870554670, −2.31676855308356340958263694260, 2.67296548747861602112455215304, 4.25219051623020014989640998550, 7.21832871575891314428271089902, 7.961453812093259537304129142922, 9.611144784286800274692146893400, 10.17765183474206511607747784791, 11.32031436263468872707154234936, 12.89136371949222549899684441440, 13.98433746968497289492782654017, 15.11336137720187425577559621820

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