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Properties

Label 2-60-60.59-c5-0-16
Degree $2$
Conductor $60$
Sign $0.995 - 0.0934i$
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  + (1.03 − 5.56i)2-s + (−13.3 + 8.09i)3-s + (−29.8 − 11.5i)4-s + (−55.7 − 4.70i)5-s + (31.2 + 82.4i)6-s + 92.5·7-s + (−95.0 + 154. i)8-s + (111. − 215. i)9-s + (−83.8 + 304. i)10-s + 666.·11-s + (490. − 88.1i)12-s + 717. i·13-s + (95.9 − 514. i)14-s + (779. − 388. i)15-s + (758. + 688. i)16-s − 525.·17-s + ⋯
L(s)  = 1  + (0.183 − 0.983i)2-s + (−0.854 + 0.519i)3-s + (−0.932 − 0.360i)4-s + (−0.996 − 0.0840i)5-s + (0.354 + 0.935i)6-s + 0.713·7-s + (−0.525 + 0.850i)8-s + (0.460 − 0.887i)9-s + (−0.265 + 0.964i)10-s + 1.66·11-s + (0.984 − 0.176i)12-s + 1.17i·13-s + (0.130 − 0.701i)14-s + (0.895 − 0.445i)15-s + (0.740 + 0.672i)16-s − 0.440·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0934i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.995 - 0.0934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.995 - 0.0934i$
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: \((2,\ 60,\ (\ :5/2),\ 0.995 - 0.0934i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.04022 + 0.0486953i\)
\(L(\frac12)\) \(\approx\) \(1.04022 + 0.0486953i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.03 + 5.56i)T \)
3 \( 1 + (13.3 - 8.09i)T \)
5 \( 1 + (55.7 + 4.70i)T \)
good7 \( 1 - 92.5T + 1.68e4T^{2} \)
11 \( 1 - 666.T + 1.61e5T^{2} \)
13 \( 1 - 717. iT - 3.71e5T^{2} \)
17 \( 1 + 525.T + 1.41e6T^{2} \)
19 \( 1 - 1.76e3iT - 2.47e6T^{2} \)
23 \( 1 + 2.08e3iT - 6.43e6T^{2} \)
29 \( 1 + 3.49e3iT - 2.05e7T^{2} \)
31 \( 1 - 2.60e3iT - 2.86e7T^{2} \)
37 \( 1 - 1.28e4iT - 6.93e7T^{2} \)
41 \( 1 - 1.26e4iT - 1.15e8T^{2} \)
43 \( 1 - 1.45e4T + 1.47e8T^{2} \)
47 \( 1 - 5.57e3iT - 2.29e8T^{2} \)
53 \( 1 - 1.14e4T + 4.18e8T^{2} \)
59 \( 1 - 8.48e3T + 7.14e8T^{2} \)
61 \( 1 - 6.58e3T + 8.44e8T^{2} \)
67 \( 1 - 3.90e4T + 1.35e9T^{2} \)
71 \( 1 - 2.48e3T + 1.80e9T^{2} \)
73 \( 1 + 2.20e4iT - 2.07e9T^{2} \)
79 \( 1 - 8.08e4iT - 3.07e9T^{2} \)
83 \( 1 + 2.73e4iT - 3.93e9T^{2} \)
89 \( 1 - 9.04e4iT - 5.58e9T^{2} \)
97 \( 1 + 1.59e5iT - 8.58e9T^{2} \)
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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

−14.20978051700068198647476725900, −12.36865735187856506363933097396, −11.68398551992250649466477714402, −11.11731615514915462732100988905, −9.684594149988503356588297639714, −8.517776985485706827460246216472, −6.46847581238832179974322604221, −4.60296182910616887476516817468, −3.92698156849190558757665407129, −1.23338273318244932801956447731, 0.65834036533418076524954170216, 4.04916320662578083618100979214, 5.36944892122828326272842366060, 6.82266268463217387829994546059, 7.67589340822992840305042770841, 8.953733820722348345033379949865, 10.96061794304388737356967464144, 11.92209683507693486882193736621, 12.93108149311996110307438210028, 14.25552599730832747247244226939

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