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Properties

Label 2-60-5.4-c5-0-3
Degree $2$
Conductor $60$
Sign $0.312 + 0.949i$
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  + 9i·3-s + (−53.1 + 17.4i)5-s − 222. i·7-s − 81·9-s + 682.·11-s − 412. i·13-s + (−157. − 477. i)15-s − 866. i·17-s − 668.·19-s + 1.99e3·21-s − 3.32e3i·23-s + (2.51e3 − 1.85e3i)25-s − 729i·27-s − 1.25e3·29-s − 9.70e3·31-s + ⋯
L(s)  = 1  + 0.577i·3-s + (−0.949 + 0.312i)5-s − 1.71i·7-s − 0.333·9-s + 1.70·11-s − 0.676i·13-s + (−0.180 − 0.548i)15-s − 0.726i·17-s − 0.424·19-s + 0.989·21-s − 1.31i·23-s + (0.804 − 0.593i)25-s − 0.192i·27-s − 0.276·29-s − 1.81·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.312 + 0.949i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.312 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.312 + 0.949i$
Analytic conductor: \(9.62302\)
Root analytic conductor: \(3.10210\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :5/2),\ 0.312 + 0.949i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.961096 - 0.695545i\)
\(L(\frac12)\) \(\approx\) \(0.961096 - 0.695545i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - 9iT \)
5 \( 1 + (53.1 - 17.4i)T \)
good7 \( 1 + 222. iT - 1.68e4T^{2} \)
11 \( 1 - 682.T + 1.61e5T^{2} \)
13 \( 1 + 412. iT - 3.71e5T^{2} \)
17 \( 1 + 866. iT - 1.41e6T^{2} \)
19 \( 1 + 668.T + 2.47e6T^{2} \)
23 \( 1 + 3.32e3iT - 6.43e6T^{2} \)
29 \( 1 + 1.25e3T + 2.05e7T^{2} \)
31 \( 1 + 9.70e3T + 2.86e7T^{2} \)
37 \( 1 + 8.80e3iT - 6.93e7T^{2} \)
41 \( 1 - 6.73e3T + 1.15e8T^{2} \)
43 \( 1 - 1.70e4iT - 1.47e8T^{2} \)
47 \( 1 - 1.29e4iT - 2.29e8T^{2} \)
53 \( 1 + 4.15e3iT - 4.18e8T^{2} \)
59 \( 1 - 2.84e4T + 7.14e8T^{2} \)
61 \( 1 - 1.03e4T + 8.44e8T^{2} \)
67 \( 1 + 6.59e4iT - 1.35e9T^{2} \)
71 \( 1 + 2.00e4T + 1.80e9T^{2} \)
73 \( 1 + 4.71e4iT - 2.07e9T^{2} \)
79 \( 1 - 2.10e4T + 3.07e9T^{2} \)
83 \( 1 - 9.03e4iT - 3.93e9T^{2} \)
89 \( 1 - 4.15e4T + 5.58e9T^{2} \)
97 \( 1 + 5.63e4iT - 8.58e9T^{2} \)
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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

−14.24764359595817219105099228218, −12.68709049705301329448063328250, −11.28786255967418069050842142746, −10.61472774369964008340455012725, −9.245011667965395394229597484721, −7.72019914300101056324490913005, −6.66231629324992382302253734281, −4.40739037559595870490914331261, −3.59117184142220902622242379066, −0.59512381136294294449448169094, 1.73461803056784524362756255733, 3.78917334962657340587010016382, 5.66383613103534572740869211562, 6.99663543264404894854553911034, 8.556378655855411996741759950469, 9.182496982075947146111588514882, 11.49978464881720519572270715121, 11.91455788202628247235687377309, 12.90244308720627007200944041962, 14.54038348738900886672153909826

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