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Properties

Label 2-60-20.3-c1-0-2
Degree $2$
Conductor $60$
Sign $0.546 - 0.837i$
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  + (0.0912 + 1.41i)2-s + (0.707 − 0.707i)3-s + (−1.98 + 0.257i)4-s + (1.32 + 1.80i)5-s + (1.06 + 0.933i)6-s + (−1.86 − 1.86i)7-s + (−0.544 − 2.77i)8-s − 1.00i·9-s + (−2.42 + 2.02i)10-s + 0.728i·11-s + (−1.22 + 1.58i)12-s + (−3.12 − 3.12i)13-s + (2.46 − 2.80i)14-s + (2.20 + 0.342i)15-s + (3.86 − 1.02i)16-s + (1.12 − 1.12i)17-s + ⋯
L(s)  = 1  + (0.0645 + 0.997i)2-s + (0.408 − 0.408i)3-s + (−0.991 + 0.128i)4-s + (0.590 + 0.807i)5-s + (0.433 + 0.381i)6-s + (−0.705 − 0.705i)7-s + (−0.192 − 0.981i)8-s − 0.333i·9-s + (−0.767 + 0.641i)10-s + 0.219i·11-s + (−0.352 + 0.457i)12-s + (−0.866 − 0.866i)13-s + (0.658 − 0.749i)14-s + (0.570 + 0.0885i)15-s + (0.966 − 0.255i)16-s + (0.272 − 0.272i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.546 - 0.837i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.546 - 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.546 - 0.837i$
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: \((2,\ 60,\ (\ :1/2),\ 0.546 - 0.837i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.817228 + 0.442446i\)
\(L(\frac12)\) \(\approx\) \(0.817228 + 0.442446i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0912 - 1.41i)T \)
3 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 + (-1.32 - 1.80i)T \)
good7 \( 1 + (1.86 + 1.86i)T + 7iT^{2} \)
11 \( 1 - 0.728iT - 11T^{2} \)
13 \( 1 + (3.12 + 3.12i)T + 13iT^{2} \)
17 \( 1 + (-1.12 + 1.12i)T - 17iT^{2} \)
19 \( 1 - 3.73T + 19T^{2} \)
23 \( 1 + (5.83 - 5.83i)T - 23iT^{2} \)
29 \( 1 - 2.64iT - 29T^{2} \)
31 \( 1 - 6.01iT - 31T^{2} \)
37 \( 1 + (-3.12 + 3.12i)T - 37iT^{2} \)
41 \( 1 + 4.24T + 41T^{2} \)
43 \( 1 + (-5.10 + 5.10i)T - 43iT^{2} \)
47 \( 1 + (-2.09 - 2.09i)T + 47iT^{2} \)
53 \( 1 + (-0.484 - 0.484i)T + 53iT^{2} \)
59 \( 1 - 4.92T + 59T^{2} \)
61 \( 1 - 2.31T + 61T^{2} \)
67 \( 1 + (5.10 + 5.10i)T + 67iT^{2} \)
71 \( 1 + 13.1iT - 71T^{2} \)
73 \( 1 + (-3.96 - 3.96i)T + 73iT^{2} \)
79 \( 1 + 7.11T + 79T^{2} \)
83 \( 1 + (-3.55 + 3.55i)T - 83iT^{2} \)
89 \( 1 - 1.03iT - 89T^{2} \)
97 \( 1 + (12.5 - 12.5i)T - 97iT^{2} \)
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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

−15.16712187851486138980539456248, −14.12514395374100963679250615448, −13.52446230718183543050247813021, −12.34747804584616398179878238383, −10.23931636111382693776655762026, −9.474051717135374842054123744816, −7.67344560209680778661792403648, −6.96363737989232728965577444062, −5.57950822027377763077666186519, −3.37130257409424508583511936349, 2.45183576347053094532963864216, 4.35002774704167954760368500884, 5.80193236186153991877192141527, 8.347483012465051057578954155923, 9.456896137864312631782028417290, 9.978912696771910400214745553378, 11.74302201500702253236755527125, 12.60571447260429012613425372687, 13.66455272458973499926049386285, 14.61475605772709772827622503276

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