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Properties

Label 2-60-60.59-c5-0-55
Degree $2$
Conductor $60$
Sign $-0.999 + 0.0435i$
Analytic cond. $9.62302$
Root an. cond. $3.10210$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4.16 − 3.83i)2-s + (7.28 − 13.7i)3-s + (2.66 − 31.8i)4-s + (−45.7 + 32.1i)5-s + (−22.4 − 85.2i)6-s − 213.·7-s + (−111. − 142. i)8-s + (−136. − 200. i)9-s + (−67.0 + 309. i)10-s + 602.·11-s + (−420. − 269. i)12-s − 333. i·13-s + (−887. + 816. i)14-s + (110. + 864. i)15-s + (−1.00e3 − 169. i)16-s + 831.·17-s + ⋯
L(s)  = 1  + (0.735 − 0.677i)2-s + (0.467 − 0.883i)3-s + (0.0831 − 0.996i)4-s + (−0.817 + 0.575i)5-s + (−0.254 − 0.967i)6-s − 1.64·7-s + (−0.613 − 0.789i)8-s + (−0.562 − 0.826i)9-s + (−0.212 + 0.977i)10-s + 1.50·11-s + (−0.842 − 0.539i)12-s − 0.547i·13-s + (−1.21 + 1.11i)14-s + (0.126 + 0.991i)15-s + (−0.986 − 0.165i)16-s + 0.697·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0435i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0435i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $-0.999 + 0.0435i$
Analytic conductor: \(9.62302\)
Root analytic conductor: \(3.10210\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :5/2),\ -0.999 + 0.0435i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.0383977 - 1.76456i\)
\(L(\frac12)\) \(\approx\) \(0.0383977 - 1.76456i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-4.16 + 3.83i)T \)
3 \( 1 + (-7.28 + 13.7i)T \)
5 \( 1 + (45.7 - 32.1i)T \)
good7 \( 1 + 213.T + 1.68e4T^{2} \)
11 \( 1 - 602.T + 1.61e5T^{2} \)
13 \( 1 + 333. iT - 3.71e5T^{2} \)
17 \( 1 - 831.T + 1.41e6T^{2} \)
19 \( 1 + 1.07e3iT - 2.47e6T^{2} \)
23 \( 1 + 2.99e3iT - 6.43e6T^{2} \)
29 \( 1 - 2.59e3iT - 2.05e7T^{2} \)
31 \( 1 + 2.39e3iT - 2.86e7T^{2} \)
37 \( 1 - 9.66e3iT - 6.93e7T^{2} \)
41 \( 1 + 669. iT - 1.15e8T^{2} \)
43 \( 1 - 1.84e3T + 1.47e8T^{2} \)
47 \( 1 + 1.40e4iT - 2.29e8T^{2} \)
53 \( 1 - 2.97e4T + 4.18e8T^{2} \)
59 \( 1 - 2.01e4T + 7.14e8T^{2} \)
61 \( 1 + 1.68e4T + 8.44e8T^{2} \)
67 \( 1 - 869.T + 1.35e9T^{2} \)
71 \( 1 + 3.58e4T + 1.80e9T^{2} \)
73 \( 1 + 3.71e4iT - 2.07e9T^{2} \)
79 \( 1 - 2.09e4iT - 3.07e9T^{2} \)
83 \( 1 + 1.86e4iT - 3.93e9T^{2} \)
89 \( 1 + 2.98e4iT - 5.58e9T^{2} \)
97 \( 1 - 1.69e5iT - 8.58e9T^{2} \)
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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

−13.36473317965681061302922938041, −12.41238486619608099690737696223, −11.71792710371644786239843503456, −10.18669766243817039101047322397, −8.927518590056195417920748194358, −6.98670107249169744130965422257, −6.27904843906662345377648605302, −3.75722194639909601070786066416, −2.85829360389234066341565755215, −0.63074163701585482395029287527, 3.43684115264467039546669946547, 4.09794927262472780084445705786, 5.84543977621935973975783526839, 7.30785171156014236278940924881, 8.800760299736265491265417365021, 9.625776103676068974033031469638, 11.59128042039292320379137514213, 12.51585538914494650162219571696, 13.74205371515694243082131883009, 14.74406644471681527756008272075

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