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Properties

Label 2-60-60.59-c1-0-4
Degree $2$
Conductor $60$
Sign $0.968 + 0.250i$
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  + (1.11 − 0.866i)2-s + 1.73i·3-s + (0.500 − 1.93i)4-s − 2.23·5-s + (1.49 + 1.93i)6-s + (−1.11 − 2.59i)8-s − 2.99·9-s + (−2.50 + 1.93i)10-s + (3.35 + 0.866i)12-s − 3.87i·15-s + (−3.5 − 1.93i)16-s + 4.47·17-s + (−3.35 + 2.59i)18-s + 7.74i·19-s + (−1.11 + 4.33i)20-s + ⋯
L(s)  = 1  + (0.790 − 0.612i)2-s + 0.999i·3-s + (0.250 − 0.968i)4-s − 0.999·5-s + (0.612 + 0.790i)6-s + (−0.395 − 0.918i)8-s − 0.999·9-s + (−0.790 + 0.612i)10-s + (0.968 + 0.250i)12-s − 1.00i·15-s + (−0.875 − 0.484i)16-s + 1.08·17-s + (−0.790 + 0.612i)18-s + 1.77i·19-s + (−0.250 + 0.968i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 + 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.968 + 0.250i$
Analytic conductor: \(0.479102\)
Root analytic conductor: \(0.692172\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :1/2),\ 0.968 + 0.250i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.10189 - 0.139958i\)
\(L(\frac12)\) \(\approx\) \(1.10189 - 0.139958i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.11 + 0.866i)T \)
3 \( 1 - 1.73iT \)
5 \( 1 + 2.23T \)
good7 \( 1 + 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 - 4.47T + 17T^{2} \)
19 \( 1 - 7.74iT - 19T^{2} \)
23 \( 1 + 3.46iT - 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 7.74iT - 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 - 10.3iT - 47T^{2} \)
53 \( 1 + 4.47T + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + 7.74iT - 79T^{2} \)
83 \( 1 - 3.46iT - 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−14.85454371831900041373055145068, −14.27361460814636433910119904715, −12.57248072451275670816841051627, −11.69263790237918546626704841057, −10.66466370514893032831262775356, −9.652140086505478420732511583620, −7.997428208638764822405330422004, −5.91230804043502411888542382903, −4.44720620051296434479123125516, −3.34562197376954970251902214197, 3.21612165045091494677779403163, 5.11991821723020958559461200269, 6.75590916740167205077218524161, 7.62130075998172126003502019318, 8.710604288240186882615011007122, 11.21786645245511680209390523208, 12.05063592837146378367776714785, 12.96969218899520038325655249781, 14.01906401124226464397554511700, 15.05036557585281203836587496349

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