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Properties

Label 2-60-20.7-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: $\chi_{60} (7, \cdot )$
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

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

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