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Properties

Label 2-60-3.2-c4-0-5
Degree $2$
Conductor $60$
Sign $-0.972 + 0.231i$
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  + (−2.08 − 8.75i)3-s − 11.1i·5-s − 27.4·7-s + (−72.3 + 36.5i)9-s − 29.1i·11-s − 289.·13-s + (−97.8 + 23.3i)15-s − 125. i·17-s + 38.9·19-s + (57.3 + 240. i)21-s − 893. i·23-s − 125.·25-s + (470. + 557. i)27-s + 438. i·29-s + 1.28e3·31-s + ⋯
L(s)  = 1  + (−0.231 − 0.972i)3-s − 0.447i·5-s − 0.561·7-s + (−0.892 + 0.450i)9-s − 0.240i·11-s − 1.71·13-s + (−0.435 + 0.103i)15-s − 0.433i·17-s + 0.107·19-s + (0.129 + 0.545i)21-s − 1.68i·23-s − 0.200·25-s + (0.645 + 0.764i)27-s + 0.520i·29-s + 1.33·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 + 0.231i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.972 + 0.231i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $-0.972 + 0.231i$
Analytic conductor: \(6.20219\)
Root analytic conductor: \(2.49042\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :2),\ -0.972 + 0.231i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.0853196 - 0.726723i\)
\(L(\frac12)\) \(\approx\) \(0.0853196 - 0.726723i\)
\(L(3)\) 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 \( 1 + (2.08 + 8.75i)T \)
5 \( 1 + 11.1iT \)
good7 \( 1 + 27.4T + 2.40e3T^{2} \)
11 \( 1 + 29.1iT - 1.46e4T^{2} \)
13 \( 1 + 289.T + 2.85e4T^{2} \)
17 \( 1 + 125. iT - 8.35e4T^{2} \)
19 \( 1 - 38.9T + 1.30e5T^{2} \)
23 \( 1 + 893. iT - 2.79e5T^{2} \)
29 \( 1 - 438. iT - 7.07e5T^{2} \)
31 \( 1 - 1.28e3T + 9.23e5T^{2} \)
37 \( 1 - 139.T + 1.87e6T^{2} \)
41 \( 1 + 1.99e3iT - 2.82e6T^{2} \)
43 \( 1 - 2.35e3T + 3.41e6T^{2} \)
47 \( 1 + 1.94e3iT - 4.87e6T^{2} \)
53 \( 1 + 4.48e3iT - 7.89e6T^{2} \)
59 \( 1 - 5.22e3iT - 1.21e7T^{2} \)
61 \( 1 + 5.26e3T + 1.38e7T^{2} \)
67 \( 1 + 5.22e3T + 2.01e7T^{2} \)
71 \( 1 + 5.89e3iT - 2.54e7T^{2} \)
73 \( 1 - 621.T + 2.83e7T^{2} \)
79 \( 1 - 4.99e3T + 3.89e7T^{2} \)
83 \( 1 + 3.80e3iT - 4.74e7T^{2} \)
89 \( 1 - 1.10e4iT - 6.27e7T^{2} \)
97 \( 1 - 5.99e3T + 8.85e7T^{2} \)
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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

−13.66751591873359758495839459999, −12.53872459247427066368865236222, −11.96876065049329814785379904652, −10.36983629102289781944787868519, −8.959426828594384291736527229231, −7.61391327845356095654830792353, −6.46694026769171902053434729854, −4.96225680909975686544558033425, −2.55283580525528739531391080813, −0.40354077423051073768864261133, 2.95964604872804357438603962473, 4.58139139962790911553421276534, 6.06356872559238745230310083564, 7.61605250197737578184279952841, 9.454640258253014620128923033930, 10.04856382443788833585086717390, 11.35543389707887393180690942918, 12.43138843698913223394890132662, 13.97468185423107779140745721654, 15.03512059810145901442148665642

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