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Properties

Label 2-60-20.3-c1-0-4
Degree $2$
Conductor $60$
Sign $0.994 + 0.103i$
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  + (1.35 − 0.394i)2-s + (−0.707 + 0.707i)3-s + (1.68 − 1.07i)4-s + (−1.75 + 1.38i)5-s + (−0.681 + 1.23i)6-s + (−2.47 − 2.47i)7-s + (1.87 − 2.11i)8-s − 1.00i·9-s + (−1.83 + 2.57i)10-s + 3.02i·11-s + (−0.437 + 1.95i)12-s + (0.363 + 0.363i)13-s + (−4.34 − 2.38i)14-s + (0.256 − 2.22i)15-s + (1.70 − 3.61i)16-s + (−2.36 + 2.36i)17-s + ⋯
L(s)  = 1  + (0.960 − 0.278i)2-s + (−0.408 + 0.408i)3-s + (0.844 − 0.535i)4-s + (−0.783 + 0.621i)5-s + (−0.278 + 0.505i)6-s + (−0.936 − 0.936i)7-s + (0.661 − 0.749i)8-s − 0.333i·9-s + (−0.579 + 0.814i)10-s + 0.913i·11-s + (−0.126 + 0.563i)12-s + (0.100 + 0.100i)13-s + (−1.16 − 0.638i)14-s + (0.0663 − 0.573i)15-s + (0.426 − 0.904i)16-s + (−0.573 + 0.573i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.103i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 + 0.103i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.994 + 0.103i$
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: \((2,\ 60,\ (\ :1/2),\ 0.994 + 0.103i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.13211 - 0.0588211i\)
\(L(\frac12)\) \(\approx\) \(1.13211 - 0.0588211i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.35 + 0.394i)T \)
3 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 + (1.75 - 1.38i)T \)
good7 \( 1 + (2.47 + 2.47i)T + 7iT^{2} \)
11 \( 1 - 3.02iT - 11T^{2} \)
13 \( 1 + (-0.363 - 0.363i)T + 13iT^{2} \)
17 \( 1 + (2.36 - 2.36i)T - 17iT^{2} \)
19 \( 1 - 4.95T + 19T^{2} \)
23 \( 1 + (-0.900 + 0.900i)T - 23iT^{2} \)
29 \( 1 + 3.50iT - 29T^{2} \)
31 \( 1 - 3.85iT - 31T^{2} \)
37 \( 1 + (0.363 - 0.363i)T - 37iT^{2} \)
41 \( 1 - 2.72T + 41T^{2} \)
43 \( 1 + (3.92 - 3.92i)T - 43iT^{2} \)
47 \( 1 + (5.85 + 5.85i)T + 47iT^{2} \)
53 \( 1 + (-3.14 - 3.14i)T + 53iT^{2} \)
59 \( 1 + 8.68T + 59T^{2} \)
61 \( 1 + 15.2T + 61T^{2} \)
67 \( 1 + (-3.92 - 3.92i)T + 67iT^{2} \)
71 \( 1 + 4.25iT - 71T^{2} \)
73 \( 1 + (-9.28 - 9.28i)T + 73iT^{2} \)
79 \( 1 + 0.399T + 79T^{2} \)
83 \( 1 + (-0.199 + 0.199i)T - 83iT^{2} \)
89 \( 1 + 4.28iT - 89T^{2} \)
97 \( 1 + (-6.73 + 6.73i)T - 97iT^{2} \)
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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

−15.14842204528183520260688574443, −13.97020667312586228184528747230, −12.80726443676120092771426974029, −11.75249915797600334180292304126, −10.68976027580865864514972331879, −9.847185577527207281079643655937, −7.35642846402922270814853054122, −6.42737634161125575354669114117, −4.53011808042202940201052835960, −3.38477227989812797661300606691, 3.23476554326180866955618992189, 5.12045987168770241831574206311, 6.26972344492416972489918848919, 7.66632211036182986470108667259, 9.043323684358846287950424396288, 11.18726220073307894381233053579, 12.00324800912537837407477218774, 12.85022802355210355245258644241, 13.77488434072548104553198607366, 15.36790760819187953318912256821

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