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Properties

Label 2-60-20.3-c1-0-3
Degree $2$
Conductor $60$
Sign $0.455 + 0.890i$
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  + (−1.19 − 0.760i)2-s + (0.707 − 0.707i)3-s + (0.844 + 1.81i)4-s + (0.432 − 2.19i)5-s + (−1.38 + 0.305i)6-s + (−0.611 − 0.611i)7-s + (0.371 − 2.80i)8-s − 1.00i·9-s + (−2.18 + 2.28i)10-s + 5.12i·11-s + (1.87 + 0.685i)12-s + (1.76 + 1.76i)13-s + (0.264 + 1.19i)14-s + (−1.24 − 1.85i)15-s + (−2.57 + 3.06i)16-s + (−3.76 + 3.76i)17-s + ⋯
L(s)  = 1  + (−0.843 − 0.537i)2-s + (0.408 − 0.408i)3-s + (0.422 + 0.906i)4-s + (0.193 − 0.981i)5-s + (−0.563 + 0.124i)6-s + (−0.231 − 0.231i)7-s + (0.131 − 0.991i)8-s − 0.333i·9-s + (−0.690 + 0.723i)10-s + 1.54i·11-s + (0.542 + 0.197i)12-s + (0.488 + 0.488i)13-s + (0.0706 + 0.319i)14-s + (−0.321 − 0.479i)15-s + (−0.643 + 0.765i)16-s + (−0.912 + 0.912i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.455 + 0.890i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.455 + 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.455 + 0.890i$
Analytic conductor: \(0.479102\)
Root analytic conductor: \(0.692172\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :1/2),\ 0.455 + 0.890i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.593269 - 0.362829i\)
\(L(\frac12)\) \(\approx\) \(0.593269 - 0.362829i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.19 + 0.760i)T \)
3 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 + (-0.432 + 2.19i)T \)
good7 \( 1 + (0.611 + 0.611i)T + 7iT^{2} \)
11 \( 1 - 5.12iT - 11T^{2} \)
13 \( 1 + (-1.76 - 1.76i)T + 13iT^{2} \)
17 \( 1 + (3.76 - 3.76i)T - 17iT^{2} \)
19 \( 1 - 1.22T + 19T^{2} \)
23 \( 1 + (-1.07 + 1.07i)T - 23iT^{2} \)
29 \( 1 - 0.864iT - 29T^{2} \)
31 \( 1 + 7.81iT - 31T^{2} \)
37 \( 1 + (1.76 - 1.76i)T - 37iT^{2} \)
41 \( 1 - 5.52T + 41T^{2} \)
43 \( 1 + (6.20 - 6.20i)T - 43iT^{2} \)
47 \( 1 + (2.29 + 2.29i)T + 47iT^{2} \)
53 \( 1 + (2.62 + 2.62i)T + 53iT^{2} \)
59 \( 1 - 0.528T + 59T^{2} \)
61 \( 1 - 4.98T + 61T^{2} \)
67 \( 1 + (-6.20 - 6.20i)T + 67iT^{2} \)
71 \( 1 + 8.10iT - 71T^{2} \)
73 \( 1 + (2.25 + 2.25i)T + 73iT^{2} \)
79 \( 1 + 15.9T + 79T^{2} \)
83 \( 1 + (-7.95 + 7.95i)T - 83iT^{2} \)
89 \( 1 - 7.25iT - 89T^{2} \)
97 \( 1 + (-0.793 + 0.793i)T - 97iT^{2} \)
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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

−15.07280915437925742761061930203, −13.28872046499776772258780600760, −12.72877450288727286117313601928, −11.58217129438887475327560269800, −10.01227946914180345037072447335, −9.110131533073862380999070467587, −8.026410503161687663804125371656, −6.69482606699020307559995109714, −4.22186681257837288851574594309, −1.87502238686416301347343536366, 2.99794960363558667981740078565, 5.64597234276793293522754012365, 6.89629736342839008472931529207, 8.336848530553082208476513259169, 9.352777320142061324897566286716, 10.61522855440015706435788908581, 11.32696421580510847118164097777, 13.61514662192035021469543101376, 14.33737008950654352740652010545, 15.60832160100677031397348988730

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