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Properties

Label 2-60-4.3-c6-0-3
Degree $2$
Conductor $60$
Sign $-0.996 - 0.0871i$
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  + (5.40 + 5.89i)2-s − 15.5i·3-s + (−5.57 + 63.7i)4-s − 55.9·5-s + (91.9 − 84.2i)6-s + 125. i·7-s + (−406. + 311. i)8-s − 243·9-s + (−302. − 329. i)10-s + 685. i·11-s + (993. + 86.9i)12-s − 3.31e3·13-s + (−739. + 677. i)14-s + 871. i·15-s + (−4.03e3 − 711. i)16-s − 3.07e3·17-s + ⋯
L(s)  = 1  + (0.675 + 0.737i)2-s − 0.577i·3-s + (−0.0871 + 0.996i)4-s − 0.447·5-s + (0.425 − 0.390i)6-s + 0.365i·7-s + (−0.793 + 0.608i)8-s − 0.333·9-s + (−0.302 − 0.329i)10-s + 0.515i·11-s + (0.575 + 0.0503i)12-s − 1.50·13-s + (−0.269 + 0.247i)14-s + 0.258i·15-s + (−0.984 − 0.173i)16-s − 0.625·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0871i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.996 - 0.0871i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $-0.996 - 0.0871i$
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: \((2,\ 60,\ (\ :3),\ -0.996 - 0.0871i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.0455131 + 1.04219i\)
\(L(\frac12)\) \(\approx\) \(0.0455131 + 1.04219i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-5.40 - 5.89i)T \)
3 \( 1 + 15.5iT \)
5 \( 1 + 55.9T \)
good7 \( 1 - 125. iT - 1.17e5T^{2} \)
11 \( 1 - 685. iT - 1.77e6T^{2} \)
13 \( 1 + 3.31e3T + 4.82e6T^{2} \)
17 \( 1 + 3.07e3T + 2.41e7T^{2} \)
19 \( 1 - 6.08e3iT - 4.70e7T^{2} \)
23 \( 1 - 6.97e3iT - 1.48e8T^{2} \)
29 \( 1 + 1.37e4T + 5.94e8T^{2} \)
31 \( 1 + 3.75e4iT - 8.87e8T^{2} \)
37 \( 1 - 6.43e4T + 2.56e9T^{2} \)
41 \( 1 - 2.74e3T + 4.75e9T^{2} \)
43 \( 1 - 1.00e5iT - 6.32e9T^{2} \)
47 \( 1 - 7.29e4iT - 1.07e10T^{2} \)
53 \( 1 - 9.96e4T + 2.21e10T^{2} \)
59 \( 1 - 2.37e5iT - 4.21e10T^{2} \)
61 \( 1 - 2.74e5T + 5.15e10T^{2} \)
67 \( 1 + 4.44e5iT - 9.04e10T^{2} \)
71 \( 1 - 3.24e5iT - 1.28e11T^{2} \)
73 \( 1 + 6.89e5T + 1.51e11T^{2} \)
79 \( 1 + 1.47e5iT - 2.43e11T^{2} \)
83 \( 1 - 3.55e5iT - 3.26e11T^{2} \)
89 \( 1 - 2.48e4T + 4.96e11T^{2} \)
97 \( 1 + 9.38e5T + 8.32e11T^{2} \)
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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

−14.59708542442551538396658879660, −13.23092201891526975437300972579, −12.35996659389117385264579911897, −11.51889916689357550157097652195, −9.478443596986609005545882029178, −7.972759873528347389247739146349, −7.17212614442494032372035976747, −5.76541215856214828722193763992, −4.33513153404102210310969557619, −2.49554679499393763858388541030, 0.32545910660387441553982753296, 2.63630337636311397037611973319, 4.11716064413313513535327059852, 5.20742726086260618440690858079, 6.97204993410699788215856884047, 8.877632388893030236600932527000, 10.10963194855209993693211971993, 11.07875135245795780138737248627, 12.07139302267130867325401337007, 13.25668610053997744574200459232

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