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Properties

Label 2-60-4.3-c6-0-18
Degree $2$
Conductor $60$
Sign $0.696 + 0.717i$
Analytic cond. $13.8032$
Root an. cond. $3.71527$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (7.41 − 3.00i)2-s + 15.5i·3-s + (45.9 − 44.5i)4-s + 55.9·5-s + (46.8 + 115. i)6-s − 453. i·7-s + (206. − 468. i)8-s − 243·9-s + (414. − 168. i)10-s + 742. i·11-s + (694. + 715. i)12-s + 3.33e3·13-s + (−1.36e3 − 3.36e3i)14-s + 871. i·15-s + (121. − 4.09e3i)16-s + 3.97e3·17-s + ⋯
L(s)  = 1  + (0.926 − 0.375i)2-s + 0.577i·3-s + (0.717 − 0.696i)4-s + 0.447·5-s + (0.216 + 0.535i)6-s − 1.32i·7-s + (0.403 − 0.915i)8-s − 0.333·9-s + (0.414 − 0.168i)10-s + 0.558i·11-s + (0.402 + 0.414i)12-s + 1.51·13-s + (−0.496 − 1.22i)14-s + 0.258i·15-s + (0.0296 − 0.999i)16-s + 0.808·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.696 + 0.717i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.696 + 0.717i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.696 + 0.717i$
Analytic conductor: \(13.8032\)
Root analytic conductor: \(3.71527\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :3),\ 0.696 + 0.717i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(3.22181 - 1.36254i\)
\(L(\frac12)\) \(\approx\) \(3.22181 - 1.36254i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-7.41 + 3.00i)T \)
3 \( 1 - 15.5iT \)
5 \( 1 - 55.9T \)
good7 \( 1 + 453. iT - 1.17e5T^{2} \)
11 \( 1 - 742. iT - 1.77e6T^{2} \)
13 \( 1 - 3.33e3T + 4.82e6T^{2} \)
17 \( 1 - 3.97e3T + 2.41e7T^{2} \)
19 \( 1 + 6.35e3iT - 4.70e7T^{2} \)
23 \( 1 - 3.26e3iT - 1.48e8T^{2} \)
29 \( 1 + 4.15e4T + 5.94e8T^{2} \)
31 \( 1 - 2.74e4iT - 8.87e8T^{2} \)
37 \( 1 + 2.70e4T + 2.56e9T^{2} \)
41 \( 1 - 9.90e4T + 4.75e9T^{2} \)
43 \( 1 - 8.20e4iT - 6.32e9T^{2} \)
47 \( 1 - 7.96e4iT - 1.07e10T^{2} \)
53 \( 1 + 1.69e5T + 2.21e10T^{2} \)
59 \( 1 - 1.52e5iT - 4.21e10T^{2} \)
61 \( 1 + 3.29e5T + 5.15e10T^{2} \)
67 \( 1 + 2.09e4iT - 9.04e10T^{2} \)
71 \( 1 - 5.90e5iT - 1.28e11T^{2} \)
73 \( 1 + 3.41e5T + 1.51e11T^{2} \)
79 \( 1 + 7.56e5iT - 2.43e11T^{2} \)
83 \( 1 - 5.45e5iT - 3.26e11T^{2} \)
89 \( 1 - 1.09e6T + 4.96e11T^{2} \)
97 \( 1 + 9.77e5T + 8.32e11T^{2} \)
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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

−13.69957577654918920247590437331, −12.85358137839272750550120889136, −11.22076823100244633270328095601, −10.55138729173839657128156339072, −9.411964066032151088226993928955, −7.34823075085535405474544712368, −5.94909161240175573577313955585, −4.50590541298579457057885231123, −3.37111445116461073963094557195, −1.27620557243834651593117189920, 1.89174399829821794576486262514, 3.44490480452893700956658382244, 5.64871033181821202020911841434, 6.10634699741010501076043594312, 7.85709445588168676571045608093, 8.954793403212620165620195171642, 10.99208203505820383471764287095, 12.05338389723371495346137598929, 12.96536493297583414427306135127, 13.91525230191904308782465895150

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