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Properties

Label 2-60-5.4-c5-0-5
Degree $2$
Conductor $60$
Sign $-0.931 + 0.364i$
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 + (−20.3 − 52.0i)5-s + 121. i·7-s − 81·9-s − 699.·11-s − 436. i·13-s + (468. − 183. i)15-s − 1.61e3i·17-s − 2.77e3·19-s − 1.09e3·21-s + 1.46e3i·23-s + (−2.29e3 + 2.12e3i)25-s − 729i·27-s − 2.23e3·29-s + 4.63e3·31-s + ⋯
L(s)  = 1  + 0.577i·3-s + (−0.364 − 0.931i)5-s + 0.937i·7-s − 0.333·9-s − 1.74·11-s − 0.716i·13-s + (0.537 − 0.210i)15-s − 1.35i·17-s − 1.76·19-s − 0.541·21-s + 0.579i·23-s + (−0.734 + 0.678i)25-s − 0.192i·27-s − 0.493·29-s + 0.865·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.931 + 0.364i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.931 + 0.364i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $-0.931 + 0.364i$
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: \((2,\ 60,\ (\ :5/2),\ -0.931 + 0.364i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.0192251 - 0.101918i\)
\(L(\frac12)\) \(\approx\) \(0.0192251 - 0.101918i\)
\(L(\frac{7}{2})\) 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 \( 1 + (20.3 + 52.0i)T \)
good7 \( 1 - 121. iT - 1.68e4T^{2} \)
11 \( 1 + 699.T + 1.61e5T^{2} \)
13 \( 1 + 436. iT - 3.71e5T^{2} \)
17 \( 1 + 1.61e3iT - 1.41e6T^{2} \)
19 \( 1 + 2.77e3T + 2.47e6T^{2} \)
23 \( 1 - 1.46e3iT - 6.43e6T^{2} \)
29 \( 1 + 2.23e3T + 2.05e7T^{2} \)
31 \( 1 - 4.63e3T + 2.86e7T^{2} \)
37 \( 1 - 5.10e3iT - 6.93e7T^{2} \)
41 \( 1 - 7.68e3T + 1.15e8T^{2} \)
43 \( 1 - 2.04e4iT - 1.47e8T^{2} \)
47 \( 1 + 2.10e4iT - 2.29e8T^{2} \)
53 \( 1 - 6.03e3iT - 4.18e8T^{2} \)
59 \( 1 - 2.77e4T + 7.14e8T^{2} \)
61 \( 1 + 4.87e4T + 8.44e8T^{2} \)
67 \( 1 + 1.12e4iT - 1.35e9T^{2} \)
71 \( 1 + 1.53e4T + 1.80e9T^{2} \)
73 \( 1 - 2.44e4iT - 2.07e9T^{2} \)
79 \( 1 + 2.37e4T + 3.07e9T^{2} \)
83 \( 1 + 1.89e4iT - 3.93e9T^{2} \)
89 \( 1 + 5.75e4T + 5.58e9T^{2} \)
97 \( 1 + 1.73e5iT - 8.58e9T^{2} \)
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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

−13.37312730972313459805017433753, −12.53479460760003439881477685633, −11.32317356628645119516890970495, −10.03874349214579017810214744948, −8.795418853035953609317336866384, −7.86601093309959708009523148346, −5.64077958887853367515789834866, −4.70679257414635754484704579280, −2.69696280274648974163018914121, −0.04559270742026217067887345557, 2.33703431379062242017320316041, 4.12582078518371921019334462799, 6.20981062787582972839024388582, 7.34660930267184447639779965043, 8.323720005750798792470212652419, 10.45141340174398342572109791451, 10.85747420716602449840995905706, 12.50091431557592048226217671860, 13.43068102749297301570317069980, 14.52062851393749130488248342667

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