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Properties

Label 2-60-15.14-c6-0-5
Degree $2$
Conductor $60$
Sign $0.877 + 0.479i$
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  + (−24.4 − 11.3i)3-s + (−124. − 8.26i)5-s + 577. i·7-s + (471. + 556. i)9-s − 1.96e3i·11-s − 2.09e3i·13-s + (2.96e3 + 1.61e3i)15-s + 1.15e3·17-s + 5.61e3·19-s + (6.55e3 − 1.41e4i)21-s + 3.38e3·23-s + (1.54e4 + 2.06e3i)25-s + (−5.22e3 − 1.89e4i)27-s + 5.72e3i·29-s + 3.69e4·31-s + ⋯
L(s)  = 1  + (−0.907 − 0.420i)3-s + (−0.997 − 0.0661i)5-s + 1.68i·7-s + (0.646 + 0.763i)9-s − 1.47i·11-s − 0.951i·13-s + (0.877 + 0.479i)15-s + 0.234·17-s + 0.818·19-s + (0.707 − 1.52i)21-s + 0.278·23-s + (0.991 + 0.131i)25-s + (−0.265 − 0.964i)27-s + 0.234i·29-s + 1.23·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.877 + 0.479i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.877 + 0.479i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.877 + 0.479i$
Analytic conductor: \(13.8032\)
Root analytic conductor: \(3.71527\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :3),\ 0.877 + 0.479i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.961101 - 0.245519i\)
\(L(\frac12)\) \(\approx\) \(0.961101 - 0.245519i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (24.4 + 11.3i)T \)
5 \( 1 + (124. + 8.26i)T \)
good7 \( 1 - 577. iT - 1.17e5T^{2} \)
11 \( 1 + 1.96e3iT - 1.77e6T^{2} \)
13 \( 1 + 2.09e3iT - 4.82e6T^{2} \)
17 \( 1 - 1.15e3T + 2.41e7T^{2} \)
19 \( 1 - 5.61e3T + 4.70e7T^{2} \)
23 \( 1 - 3.38e3T + 1.48e8T^{2} \)
29 \( 1 - 5.72e3iT - 5.94e8T^{2} \)
31 \( 1 - 3.69e4T + 8.87e8T^{2} \)
37 \( 1 - 3.26e4iT - 2.56e9T^{2} \)
41 \( 1 - 5.65e4iT - 4.75e9T^{2} \)
43 \( 1 + 1.13e5iT - 6.32e9T^{2} \)
47 \( 1 - 1.57e5T + 1.07e10T^{2} \)
53 \( 1 - 1.09e5T + 2.21e10T^{2} \)
59 \( 1 - 2.27e5iT - 4.21e10T^{2} \)
61 \( 1 + 2.71e5T + 5.15e10T^{2} \)
67 \( 1 + 2.19e5iT - 9.04e10T^{2} \)
71 \( 1 + 5.22e5iT - 1.28e11T^{2} \)
73 \( 1 + 1.78e3iT - 1.51e11T^{2} \)
79 \( 1 - 7.94e5T + 2.43e11T^{2} \)
83 \( 1 - 2.13e5T + 3.26e11T^{2} \)
89 \( 1 + 1.24e6iT - 4.96e11T^{2} \)
97 \( 1 + 4.92e5iT - 8.32e11T^{2} \)
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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

−13.52643686443001705932759551102, −12.21799241476447878074882689087, −11.77444590027629546353744831838, −10.65830262606501016901231975867, −8.791609893555820590958908829074, −7.79005359586298991499247787316, −6.10441612602096374228991878249, −5.17399548066224772706649830281, −3.02429099279854825434430251887, −0.69971005625398283177167518086, 0.894644722664201576105466403979, 3.94306864775035807573592019936, 4.67557877410092522445666941942, 6.83361978110350281082075341510, 7.51962200629903169454262827004, 9.614179752549905148719400652182, 10.58347122095072586566711930535, 11.59164339158612048522232054138, 12.53175117966271098839194554958, 14.00638927076711252132808070680

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