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Properties

Label 2-60-15.14-c4-0-3
Degree $2$
Conductor $60$
Sign $0.453 - 0.891i$
Analytic cond. $6.20219$
Root an. cond. $2.49042$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (8.97 + 0.697i)3-s + (−9.58 + 23.0i)5-s + 67.4i·7-s + (80.0 + 12.5i)9-s − 155. i·11-s + 206. i·13-s + (−102. + 200. i)15-s + 119.·17-s + 492.·19-s + (−47.0 + 605. i)21-s − 353.·23-s + (−441. − 442. i)25-s + (709. + 168. i)27-s − 917. i·29-s − 632.·31-s + ⋯
L(s)  = 1  + (0.996 + 0.0774i)3-s + (−0.383 + 0.923i)5-s + 1.37i·7-s + (0.987 + 0.154i)9-s − 1.28i·11-s + 1.22i·13-s + (−0.453 + 0.891i)15-s + 0.413·17-s + 1.36·19-s + (−0.106 + 1.37i)21-s − 0.667·23-s + (−0.705 − 0.708i)25-s + (0.973 + 0.230i)27-s − 1.09i·29-s − 0.657·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.453 - 0.891i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.453 - 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.453 - 0.891i$
Analytic conductor: \(6.20219\)
Root analytic conductor: \(2.49042\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :2),\ 0.453 - 0.891i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.72958 + 1.05999i\)
\(L(\frac12)\) \(\approx\) \(1.72958 + 1.05999i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-8.97 - 0.697i)T \)
5 \( 1 + (9.58 - 23.0i)T \)
good7 \( 1 - 67.4iT - 2.40e3T^{2} \)
11 \( 1 + 155. iT - 1.46e4T^{2} \)
13 \( 1 - 206. iT - 2.85e4T^{2} \)
17 \( 1 - 119.T + 8.35e4T^{2} \)
19 \( 1 - 492.T + 1.30e5T^{2} \)
23 \( 1 + 353.T + 2.79e5T^{2} \)
29 \( 1 + 917. iT - 7.07e5T^{2} \)
31 \( 1 + 632.T + 9.23e5T^{2} \)
37 \( 1 + 1.75e3iT - 1.87e6T^{2} \)
41 \( 1 + 1.44e3iT - 2.82e6T^{2} \)
43 \( 1 - 627. iT - 3.41e6T^{2} \)
47 \( 1 - 3.70e3T + 4.87e6T^{2} \)
53 \( 1 + 3.66e3T + 7.89e6T^{2} \)
59 \( 1 + 1.42e3iT - 1.21e7T^{2} \)
61 \( 1 - 1.73e3T + 1.38e7T^{2} \)
67 \( 1 - 6.45e3iT - 2.01e7T^{2} \)
71 \( 1 - 2.43e3iT - 2.54e7T^{2} \)
73 \( 1 + 2.69e3iT - 2.83e7T^{2} \)
79 \( 1 - 3.21e3T + 3.89e7T^{2} \)
83 \( 1 - 1.51e3T + 4.74e7T^{2} \)
89 \( 1 - 7.46e3iT - 6.27e7T^{2} \)
97 \( 1 + 1.07e4iT - 8.85e7T^{2} \)
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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

−14.38388977364663061667626361099, −13.85347274368264306387219258645, −12.17448842988640731296679755516, −11.21643307242294141485196744334, −9.634770509183780890542349751616, −8.661294154987097778037139885231, −7.45313783098253575927963578941, −5.89004960874664409436784527856, −3.68105186749132931285406220859, −2.40612228916029737128184307781, 1.19041095810324302571813162103, 3.54246442195292252087487871660, 4.84737540781714105724804748282, 7.32490947124361470185694644849, 7.940763980364289395075684627927, 9.467702180279426942611234617421, 10.36782810342882068069605844286, 12.23168270470568869925790107107, 13.09346285671287560426954118839, 14.04728545469122313467498196001

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